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A    TREATISE 


ON 


THE  THEORY  AND  SOLUTION 


07 


•ALGEBRAICAL  EQUATIONS; 


BY 


JOHN    MACNIE,   M.A 


A.    S.     BARNES     &     COMPANY, 

NEW   YORK,  CHICAGO,  &  NEW  ORLEANS. 
18/6. 


A    TREATISE 


ON 


THE  THEORY  AND  SOLUTION 


07 


ALGEBRAICAL  EQUATIONS; 


BY 


JOHN    MACNIE,   M.A 


A,     S.     BARNES     &     COMPANY, 

NEW   YORK,  CHICAGO,  &  NEW  ORLEANS. 
I876. 


<4*  * 




COPYRIGHT,  1875,  by  JOHN  MACNIE. 


PREFACE. 


may  therefore  be  easily  read  by  those  who  have  passed  through 
the  usual  course  in  algebra. 

The  work  had  its  first  inception  in  some  vain  attempts  at 
so  modifying  Sturm's  Method  as  to  lessen  the  great  labor 
attending  an  analysis  by  that  method  of  most  equations  of 
above  the  fourth  degree.  Becoming  convinced  that,  from  the 
nature  of  the  case,  no  such  modification  is  possible  that  does 


ERRATA 


Page   2,  line  28,   for   h   read  j-^  .£ , 


" 


8,  "  25,    for  root  read   rm/  roo^. 

19,  "       2,    for  square  read   square  root. 

19,  "  26,   for  — yCfread    +  ^1x7 

28,  "       2,    for   X  +    read  X       . 

43,  "       3,    for   Xa   read    X2. 

61,  "  11,    for  com.   dw.  read    greatest  com.  div. 

82,  "  20,    dele  Teschendorf. 

85,  "       7,   for   (yz  +  yu  +  zu)  read  (yz-f  yu  +  zu)! 

85,  "  10,   for   (y2  +  z2  +  u2)  read  (y2  +  z2  +  u2)2 

129,  "  24,   for  pairs   read  groups. 

147.  "  15,  for  (— a— B)  read  — (a— B.) 


PREFACE. 


THE  following  treatise  is  designed  to  present,  as  suc- 
cinctly as  is  consistent  with  a  due  presentation  of  the 
subject,  the  general  theory  of  algebraical  equations,  while 
special  attention  is  given  to  the  analysis  and  solution  of 
equations  with  numerical  coefficients. 

The  work  may  be  regarded  as  a  complement  to  the  more 
advanced  treatises  on  algebra,  in  which  the  general  theory 
of  equations,  if  discussed  at  all,  is  necessarily  compressed 
into  a  space  altogether  inadequate  to  a  satisfactory  exposition 
of  this  important  branch  of  mathematical  study.  In  order 
to  preseiwe  this  character  of  the  work  as  a  sequel  to  ordinary 
algebra,  algebraical  methods  have  been  carefully  adhered  to 
throughout,  except  in  a  few  articles,  in  which  trigonometrical 
expressions  have  unavoidably  been  introduced.  The  treatise 
may  therefore  be  easily  read  by  those  who  have  passed  through 
the  usual  course  in  algebra. 

The  work  had  its  first  inception  in  some  vain  attempts  at 
so  modifying  Sturm's  Method  as  to  lessen  the  great  labor 
attending  an  analysis  by  that  method  of  most  equations  of 
above  the  fourth  degree.  Becoming  convinced  that,  from  the 
nature  of  the  case,  no  such  modification  is  possible  that  does 


iy  PREFACE. 

not  impair  the  certainty  which  constitutes  the  chief  excellency 
of  the  method,  the  author  turned  his  attention  to  an  investi- 
gation of  the  possibility  of  a  satisfactory  analysis  by  means  of 
Fourier's  Theorem.  This  he  soon  saw  could  be  effected  if, 
by  any  means,  the  presence  of  imaginary  roots  in  a  given 
interval  could  be  readily  ascertained.  The  results  of  this 
investigation  are  given  (Chap.  X)  in  a  method  of  analysis 
based  upon  Fourier's  Theorem,  a  method  that  possesses  the 
merit  of  at  least  great  facility  as  compared  with  that  of 
Sturm. 

The  method  involves,  however,  an  extension  of  the  applica- 
tion of  Horner's  Method,  especially  a  generalization  of  the 
principle  of  trial  divisors,  not  given  in  the  great  majority  of 
treatises  upon  equations.  This  first  suggested  the  idea  of 
writing  a  short  account  of  Homers  Method  in  which  its 
capabilities  should  be  fully  exhibited,  with  the  method  of 
analysis  based  on  that  method  in  conjunction  with  Fourier's 
Theorem.  A  natural  extension  of  that  idea  led  to  the  present 
treatise,  the  first  upon  this  subject  that,  as  far  as  the  author 
has  been  able  to  ascertain,  has  been  published  in  the  United 
States. 

The  treatise,  though  kept  within  its  present  dimensions  by 
the  exclusion  of  much  that  had  been  prepared,  will  be  found 
to  contain  all  the  propositions  generally  given  in  an  elemen- 
tary treatise  upon  this  subject,  with  a  few  exceptions.  Thus 
Newton's  method  of  approximation,  and  that  of  Lagrange, 
have  been  excluded,  as  being  entirely  superseded  by  that  of 
Horner,  to  which,  even  in  the  most  favorable  cases,  they  are 
inferior  in  symmetry,  compactness,  and  facility.  The  theory 
of  determinants  has  not  been  introduced,  on  the  ground  that 
a  suitable  account  would  require  to  itself  a  volume  of  the 
dimensions  of  the  present  treatise.  The  student  desirous  of 
information  upon  the  foregoing  and  other  omitted  topics  may 


PREFACE.  V 

consult  Todhunter's  excellent  treatise,  the  best,  upon  the 
whole,  with  which  the  author  is  acquainted. 

On  account  of  the  interest  naturally  attaching  to  the  sub- 
ject, the  algebraical  solution  of  equations  has  (Chap.  VII) 
been  treated  of  at  some  length.  In  Art.  121,  133-146,  are 
given  some  results  that  may  be  found  of  interest,  and  which, 
it  is  believed,  are  now  published  for  the  first  time. 

A  chapter  (VIII)  has  been  devoted  to  Sturm's  Theorem, 
thus  enabling  the  student  to  form  for  himself  a  clear  estimate 
of  the  peculiar  excellencies  and  disadvantages  of  the  method 
of  analysis  based  upon  that  theorem,  and  to  institute  a  fair 
comparison  between  it  and  the  method  explained  in  Chap.  X. 

A  short  chapter  on  cubic  equations  has  been  inserted,  in 
which  a  method  of  procedure  is  given  that  relieves  the  solu- 
tion of  the  cubic  from  much  of  its  tentative  character,  and 
reduces  the  arithmetical  labor  to  a  minimum. 

To  render  the  treatise  more  convenient  for  the  work  of  the 
class-room,  the  subject-matter  has  been  thrown,  as  far  as  pos- 
sible, into  the  form  of  propositions  with  their  dependent 
corollaries.  The  number  of  exercises  given  will,  it  is  hoped, 
be  found  sufficient  in  number  to  illustrate  every  part  of  the 
subject  and  not  so  difficult  as  to  needlessly  consume  time. 

JOHN  MACNIE. 
Newburgh,  August,  1875. 


CONTENTS 


INTRODUCTION. 

ART.  PAGE 

3.  Definition  of  a  function.    4.  Derived  functions 3-4 

CHAPTER  I. 

FUNDAMENTAL   PROPERTIES   OF   EQUATIONS. 

6.  Any  term  in  a  rational  integral  function  may  be  made  greater 
than  the  sum  of  all  the  terms  that  follow  it ;  or,  that  precede  it .  .       5 

7.  To  determine  the  form  of  /(a*)  when  x  +  y  is  put  for  a; C 

8.  A  function  f(x)  will  vary  continuously  from  /(«)  to  /(&)  if  x 
vary  continuously  from  a  to  b 7 

9.  A  root  of  fix)  =  0  must  lie  between  a  and  b  if  f{a)  and  f(b) 
differ  in  sign 8 

11.  Every  equation  of  an  odd  degree  has  at  least  one  real  root 9 

12.  Every  equation  of  an  even  degree  with  its  final  term  negative 
has  at  least  two  real  roots 9 

14.  If  a  function  fix),  and  the  successive  quotients  arising  from  the 
division,  be  divided  by  x  —  a,  the  successive  remainders  will  be 
f(a),  ftal  iMa),.. .  .£/.(«) 10 

15.  If  a  is  a  root  of  f(x)  =  0,  then  x  —  a  is  a  factor  of  f(x) ;  and,. .     11 

16.  Conversely,  a  is  a  root  of  f(x)  =  0,  if  x  —  a  divide  f(x)  exactly..     11 
19.  To  find  the  quotient  and  remainder  when  f(x)  is  divided  by  x— a,     12 

CHAPTER   II. 

IMAGINARY   EXPRESSIONS.      CAUCHY'S   THEOREM. 

25.  The  sum,  difference,  product,  or  quotient,  of  expressions  of  the 

form  A  +  B\/—l  have  the  same  form 17 

27.  Conjugate  expressions.    28.  Modulus  of  conjugate  expressions. .  18 

31.  Powers  of  \/~-\.    32.  Square  root  of  /v/— 1^ 19 

33.  Each  of  the  equations  xn  =  ±  1,  a"1  =  ±  <\/ — 1  has  a  root 19 

34.  Every  rational  integral  equation  has  a  root,  real  or  imaginary.  . .  20 

35.  If  a  +  b%/— 1  is  a  root  of  f(x)  =  0,  a  and  b  must  be  finite  quan- 
tities   22 


CONTEXTS.  Ml 


CHAPTER  III. 

GENERAL   PROPERTIES   OF  EQUATIONS 

dependent  on  the  principle  tliat  every  equation  has  a  root. 

ART.  PAGE 

:J6.  An  equation  of  the  nth  degree  has  n  roots,  and  no  more 24 

40.  A  function  f{x)  of  the  nth  degree  has  *("-*>  ••(»-r+1>  factors  of 

the  r1'1  degree 26 

40.  Imaginary  roots  occur  in  conjugate  pairs 27 

45^  The  coefficients  of  an  equation  are  certain  functions  of  the 
roots 29 

48.  An  equation  having  unity  as  its  leading  coefficient,  and  the  re- 
maining coefficients  integral,  cannot  have  a  rational  fractional 
root 32 

50.  In  such  an  equation  the  rational  roots  are  integral  factors  of  the 
final  term  3:5 

53.  Descartes'  Rule  of  Signs 34 


CHAPTER  IV.     . 

TRANSFORMATION   OF    EQUATIONS. 

57.  To  transform  an  equation  having  negative  or  fractional  expo- 
nents into  another  having  only  positive  integral  exponents :><> 

50.  To  transform  an  equation  into  another  having  its  roots  m  times 
as  great  38 

61.  To  transform  an  equation  so  as  to  change  the  signs  of  its  roots.  .     40 

04.  To  transform  an  equation  into  another  having  as  roots  the  re- 
ciprocals of  the  roots  of  the  proposed  equation 41 

G6.  To  transform  an  equation  into  another  having  as  roots  the 
squares  of  the  roots  of  the  proposed  equation 42 

68.  To  transform  an  equation  into  another  having  as  roots  those  of 
the  proposed  equation,  each  increased  or  diminished  by  a  given 
quantity 44 

71.  To  transform  an  equation  into  another  in  which  any  assigned 
term  shall  be  absent 47 

CHAPTER  V. 

LIMITS   OF  THE   ROOTS   OF  EQUATIONS. 

74.  If  —  k  be  the  greatest  negative  coefficient,  then  n  +  1  is  a  supe- 
rior limit  to  the  positive  roots 49 

75  If  —  k  be  the  greatest  negative  coefficient,  and  a"l-r  the  highest 
power  of  x  with  a  negative  coefficient,  then  <y//c  +1  is  a  supe- 
rior limit 50 


Mil  CONTENTS. 

ART.  PAGE 

70.  The  greatest  quotient  obtained  by  dividing  any  negative  coeffi- 
cient by  the  sum  of  all  the  positive  coefficients  that  precede 
it  is,  when  increased  by  unity,  a  superior  limit  to  the  positive 
roots 50 

80.  An  odd  number  of  the  roots  of  f(x)  =  0  lies  between  a  and  b, 

if  f(a)  and  f(b)  differ  in  sign 54 

82.  A  real  root  of  f\(x)  =  0  lies  between  every  adjacent  two  of  the 
real  roots  of  f(z)  =  0 55 

84.  If  f(x)  —  0  has  r  roots,  each  equal  to  a,  then  f\  (x)  =  0  has 
r  —  1  roots,  each  equal  to  a 57 

87.  If '  any  derived  equation  fr  (x)  =  0  has  p  imaginary  roots, 
f(x)  =  0  must  have  at  least  as  many 58 


CHAPTER  VI. 

DEPRESSION   OF   EQUATION'S. 

92.  An  equation  f(x)  =  0  has,  or  has  not,  equal  roots  according 
as  f{x)  and  f\(x)  have,  or  have  not,  a  common  measure  in- 
volving x 60 

94.  Method  for  determining  the  equal  roots  of  an  equation 62 

95.  If  an  equation  have  incommensurable  equal  roots,  there  must 

be  at  least  two  pairs  of  such  roots 08 

99.  Reciprocal  Equations  are  either  of  an  even  degree  with  the 

final  term  positive,  or  may  be  reduced  to  that  form 06 

100.  A  reciprocal  equation  of  the  form  above  mentioned  may  be  de- 
pressed to  an  equation  of  one-half  its  own  degree 66 

102.  Binomial  Equations  have  their  roots  all  different 68 

103.  Any  algebraical  quantity  has  n  different  n"'  roots 68 

104.  All  these  roots  may  be  found  by  multiplying  one  of  them  by 
the  n  different  nth  roots  of  unity OS 

106.  If  a  be  a  root  of  xn  —  l  =  0,  then  any  integral  power  of  a  is  a 

root 69 

108.  If  a  be  a  root  of  xn  +  1  =  0,  then  any  odd  integral  power  of  a 

is  a  root 69 

109.  If  m  be  prime  to  n,  the  equations  x"'  —  l  =  0,  xn  —  1=  0, 
have  no  common  root  but  unity 09 

110.  If  a  is  an  imaginary  root  of  xn  —  \  =  0,  where  n  is  a  prime  num- 
ber, then  all  the  roots  are  found  in  the  series  1,  «,  a'2,. .  .a"-1  . .     09 

111.  The  solution  of  a"  —  1  =  0,  where  w  is  a  composite  number, 
may  be  made  to  depend  on  the  solution  of  the  equations 
a^  — 1  =  0,  xi  —  1  =  0,  &c,  where  p,  q,  &c,  are  the.  different 
prime  factors  of  n 70 

113.  The  solution  of  xn  ±  1  =  0  may  be  reduced  to  that  of  an  equa- 
tion of  not  more  than  one-half  the  degree  of  xn 71 


CONTENTS.  IX 

CHAPTER  VII. 

SOLUTION   OF   EQUATIONS   BY   GENERAL   FORMULAS. 

ART.  PAGK 

118.  Cardan's  Formula  for  cubic  equations 75 

121.  Investigation  showing  what  functions  of  the  roots  are  obtained 

by  Cardan's  Formula 79 

124.  Remarks  on  the  expansion  of  (m  +  n\/  —  l)  * 81 

125.  Solution  by  means  of  Trigonometrical  Tables 81 

127.  Ferrari's  Solution  of  the  biquadratic 83 

128.  Descartes'  Solution 83 

130.  Euler's  Solution 85 

132-145.  Investigation  and  illustration  of  a  general  method  for  re- 
ducing the  solution  of  an  equation  of  the  2ii"'  degree  to  that  of 

two  equations  of  the  nth  degree ......" 86 

CHAPTER  VIII. 

STURM'S    THEOREM. 
146-160.  Full  account  of  this  method  of  analysis 96-105 

CHAPTER  IX. 

HORNER'S    METHOD. 

161.  General  view  of  the  method 106 

164-166.  Theory  of  trial  divisors  for  approximating  to  a  single  root.  108 

170.  Method  of  contraction 114 

175.  Theory  of  trial  divisors  for  approximating  to  several  roots 118 

CHAPTER  X. 

ANALYSIS   OF  EQUATIONS   BY   FOURIER'S  THEOREM. 

179.  Fourier's  Theorem 126 

180.  If  f(x)  =  0  has  m  more  variations  than  f{a  +  x')  —  0,  then 
f(x)  =  0  has  m  roots  in  the  interval  [0,  a],  or  m  minus  some 
even  number 128 

181.  If  no  variations  are  lost  in  the  transformation,  there  are  no 
roots  in  the  interval 128 

182.  If  an  odd  number  of  variations  is  lost,  there  is  some  odd  num- 
ber of  roots  in  the  interval 1 28 

183.  If  an  even  number  of  variations  is  lost,  there  is  either  no  root 

or  some  even  number  of  roots  in  the  interval 128 

184.  Zero  coefficients,  in  certain  cases,  indicate  imaginary  roots 128 

187.  Method  of  ascertaining  the  presence  of  incommensurable  equal 

roots 133 


X  CONTENTS. 

AKT.  PAGE 

188.  Coefficient  functions  defined 134 

189.  There  are  at  least  as  many  imaginary  roots  in  an  equation  as 
there  are  variations  in  the  signs  of  its  coefficient  functions 135 

T  +  1 

100.  If  Cr  <  — ■ — Cr+i.d—i,  a  pair  of  imaginary  roots  is  indi- 
cated     135 

102-194.  If  an  equation  have  a  pair  of  imaginary  roots  a±/3y/— 1, 
then  some  transformed  equation  f(a  -f  cc')  =  0  will  give  indi- 
cations of  their  presence 13G 

CHAPTER  XI. 

CUBIC    EQUATIONS. 

202.  In  the  equation  x^  —  qx  —  r  =  0,  the  greatest  root  Xi  lies  be- 
tween 'v/fg  and  \/4i* 148 

204.  Formulae  for  the  remaining  roots  in  terms  of  X\ 149 

CHAPTER  XII. 

SYMMETRICAL   FUNCTIONS   OF  THE   ROOTS. 

207.  To  obtain  the  sum  of  the  ma  powers  of  the  roots  in  terms  of 

the  coefficients  and  inferior  powers 15G 

211.  Any  rational  symmetrical  function  of  the  roots  of  an  equation 
can  be  expressed  in  terms  of  the  coefficients  and  functions  of 
lower  order 150 

214.  To  obtain  the  equation  of  the  squares  of  the  differences  of  the 

roots  of  a  proposed  equation 160 

21G-219.  Application  of  symmetrical  f mictions  to  the  determination 

of  the  roots 102 

220,  221.  Determination  of  the  values  of  imaginary  roots 164 

CHAPTER  XIII. 

ELIMINATION. 

227.  Elimination  by  means  of  symmetrical  functions 169 

229.  On  the  degree  of  the  final  equation 170 

231.  Elimination  by  the  process  for  the  greatest  common  measure.  .  171 

232.  Improved  method  of  elimination 171 

ANSWERS 177 


THEOEY  AND   SOLUTION 


OF 


ALGEBRAICAL     EQUATIONS 


INTRODUCTION. 


1.  Algebraical  equations  of  the  first  and  second  degrees  are 
generally  fully  treated  of  in  elementary  works  on  algebra; 
the  student  may  therefore  be  supposed  to  have  a  knowledge 
of  the  subject  so  far,  and  to  be  acquainted  with  the  meaning 
of  the  terms,  equation,  root,  member,  &c.  The  present  treatise 
will  be  devoted  to  a  discussion  of  the  general  theory  of  alge- 
braical equations,  and  the  methods  of  solution  for  equations 
of  the  third  and  higher  degrees,  sometimes  called  the  Higher 
Equations. 

We  shall  generally  express  an  equation  of  the  nth  degree 
under  the  form, 


Cnxn  +  CUz"-1  + 


C2x2  +  Cyx  +  C0  =  0, 


in  which  Cr,  the  coefficient  of  the  rth  power  of  x,  is  a  known 
quantity,  positive,  negative,  or  zero.  The  final  term,  C0 ,  which 
may  be  regarded  as  the  coefficient  of  .t°,  is  frequently  called 
the  absolute  or  independent  term. 

The  equation  is  said  to  be  complete  when  it  contains  all 
the  powers  of  x  from  xn  to  x°,  otherwise  it  is  incomplete. 

When  n,  the  exponent  of  the  highest  power  of  x,  is  a 
positive  integer,  the  equation  is  said  to  be  rational  and  inte- 
gral, that  is,  x  does  not  occur  with  fractional  or  negative 
exponents. 


2  ALGEBRAICAL    EQUATIONS. 

Thus,    5s6  +  17s5  —  39s3  +  106a;2  —  53s  —  17  =  0 
is  a  rational  integral  equation ;  while 

X5  _  23^     +  5s3  +  6s"3*     _  7  —  0, 
or  s5  —  2SV&+  5#  +  6^—  7=0, 

is  an  irrational  equation ;  and 

7s4  —  3s*  +  9s~2  +  13s-4  +  8=0, 

is  both  irrational  and  fractional. 

Since  irrational  and  fractional  equations  may  always  be 
reduced  to  a  rational  and  integral  form  (see  Art.  57),  the 
equations  treated  of  in  this  work  will  always  be  supposed  to  be 
rational  and  integral. 

By  dividing  both  members  of  an  equation  by  the  coefficient 
of  the  highest  power  of  s,  we  obtain  it  under  the  form, 

xn  -f  6V1Z"-1  +.....  C2x2  +  C\x  +  C0  =  0, 

a  form   which  will  sometimes  be  found  advantageous  when 
enunciating  some  of  the  properties  of  equations. 

2.  Any  quantity,  real  or  imaginary,  which,  when  substi- 
tuted for  x  in  the  expression, 

Cnxn  4-  CUa*"1  +    .    .    .    .   C2x2  +  &x  +  C0, 

causes  the  whole  to  become  zero,  is  said  to  be  a  root  of  the 
equation, 

Cnxn  +  C^s""1  +    •    •    .    •    C2x2  +  Cxx  +  C0  =  0. 

The  solution  of  an  equation  consists  in  the  determination 
of  these  roots. 

We  know  that  for  the  quadratic  equation, 

C2x2  +  C\x  +  C0  =  0, 

the  roots  are  expressed  by  the  formula, 

x  =  t.(-G±  VW-WM). 

This  formula  contains  the  algebraical  solution  of  the  equa- 
tion of  the  second  degree,  in  terms  of  the  general  symbols 
C2 ,  Ciy  C0.  The  numerical  solution  of  any  given  equation 
of  that  degree  we  may  obtain  by  substituting  in  the  formula 


INTRODUCTION.  3 

the  known  values  of  the  coefficients.  To  obtain  for  the  higher 
equations  similar  formulas,  expressing  the  roots  in  terms  of  the 
eoefficients,  was  long  considered  the  most  important  problem 
in  algebra.  But,  as  will  hereafter  be  seen,  no  such  formula 
has  been  found  for  equations  beyond  the  fourth  degree,  and 
even  those  that  have  been  obtained  for  the  cubic  and  bi- 
quadratic are  not  always  available  for  purposes  of  numerical 
computation. 

Yet,  though  the  search  for  general  formulas,  to  express  the 
roots  of  equations  of  the  fifth  and  higher  degrees,  has  been 
entirely  unsuccessful,  we  are  in  possession  of  methods  by  which 
we  can  obtain  exactly,  when  commensurable,  or  approximate  to 
when  incommensurable,  any  root  of  a  numerical  equation,  that 
is,  of  an  equation  having  its  coefficients  known  numbers. 
These  methods  depend  on  the  general  properties  of  equations, 
which  we  shall  first  demonstrate  as  a  necessary  preliminary  to 
the  study  of  methods  of  numerical  solution. 

3.  An  algebraical  expression  involving  x  is  often  conven- 
iently referred  to  as  a  function  of  x,  which  may  be  defined  as 
follows  : 

A  Function  of  x  is  an  expression  that  depends  for  its 
value  on  x. 

Thus,  x2,  \/x,  ax2  +  lx  +  c,  5x*—  1x  +  32,  are  functions 
of  x,  since  their  values  depend  on  that  of  x. 

The  symbol  for  a  function  is  one  of  the  letters  /,  F,  0,  &c.. 
prefixed  to  the  symbol  of  the  variable  quantity.  Thus,  f(x), 
F(z),  fi(x),  (p(x),  /(?/),  f(z  +  y),  &c,  may  be  used  to  repre- 
sent any  expression  involving  x  or  y,  or  both  x  and  y,  it 
being  of  course  understood  that,  in  the  same  investigation, 
a  particular  symbol,  f(x)  for  example,  shall  always  denote 
the  same  expression  :  thus,  if 

f{x)     =  5x*  -  3Z3  +  2z2  +  1,  then 

f(a)     =  5a4  -  3«3  +  2«2  +  1, 

/(0)      =0  -0  +0  +1  =  1, 

/(I)     =  5  -  3  +2  +1  =  5, 

/(-2)  =  5  x  (-2)4- 3  x  (-2)3+ 2  x  (-2)2+1  =  113, 
/(3)      =  5  x  34       -  3  x  33       +  2  x  S2       +  1  =  343, 


4  ALGEBRAICAL    EQUATIONS. 

that  is,  if/  (a)  denote  a  certain  expression  involving  x,  then 
f(a),  /(0),  /(3),  &c.,  denote  the  same  expression,  or  its 
numerical  value,  when  «,  0,  3,  &c,  are  substituted  for  a. 

4.  The  symbols,  f(x),  fix),  .  .  .  /4(a),  &c,  are  often  em- 
ployed to  denote  functions  derived  from  a  primitive  function, 
/(a),  according  to  some  given  law :  thus,  if 

f(x)  =  CtX*  +  C^x1  +  Ctxl  +  dx  +  (70,  then 

f(x)  =  4<74a3  +  W3X*  +  %Otx  +  C\, 

/2(a)   =  4-3  CW  +  3-2-6'3x  +  2C2, 

/3(a)  =  4-3-2-C4^  +  3-2-(73, 

f,{x)  =  4  32-  C4. 

In  this  example  the  law  of  derivation  is  that  each  function, 
after  the  first,  is  derived  from  that  preceding,  by  multiplying 
each  term  by  the  exponent  of  x  in  that  term,  and  then  dimin- 
ishing that  exponent  by  unity.  When  derived  functions  are 
mentioned,  it  is  generally  functions  derived  from  a  primitive 
function,  /(a),  according  to  the  foregoing  law,  that  are  meant. 
The  function  denoted  by  fi(x)  is  called  the  first  derived  func- 
tion of  f(x);  /2(a),  again,  which  is  the  first  derived  function 
of  f\(x),  is  the  second  derived  function  of  f(x),  and  so  on, 
f(x)  denoting  the  rth  derived  function  of  /(a).  When  f(x) 
is  of  n  dimensions,  there  may  evidently  be  n  derived  functions. 
The  nth  derived  function  f„(x),  though  independent  of  x,  may 
for  convenience  be  included  among  the  derived  functions,  as 
being -tlerived  from  f(x)  by  a  repetition  of  the  process  by 
which  the  preceding  functions  were  obtained. 

EXERCISES. 

Write  out  the  derived  functions  of  each  of  the  following 
functions;  and  find  the  values  of  /(0),  /(l),  /(2),  /(3). 

1.  f(x)  =  x*  +  7a3  +  8a*  —  10a  —  13. 

2.  f{x)  =  3a?  +  13a*  -  8a3  +  18. 

3.  f(x)  =  a*  —  13a4  +  29a3  -  52a*  -  16. 

4.  /(a)   =  a8  —  8a7  -f  7a6  —  6a5  —  5a2  +  4. 

5.  Find  the  rfh  derived  function   of 

Cnx»  +  CUa""1  +    .    .    .    .    CW  +  C\x  +  C0. 


FUNDAMENTAL    PROPERTIES   OF   EQUATIONS. 


CHAPTER    I. 

FUNDAMENTAL  PROPERTIES  OF  EQUATIONS,  INDEPENDENT  OF 
THE   PRINCIPLE  THAT   EVERY   EQUATION   HAS  A  ROOT. 

5.  The  most  fundamental  proposition  respecting  equations 
is  that  proved  by  Cauchy's  Theorem,  Chap.  II,  namely,  that 
every  equation  has  a  root.  Some  important  propositions  may, 
however,  be  proved  independently  of  that  theorem ;  and 
others,  again,  are  necessary  as  an  introduction  to  it.  These 
form  the  subject  of  the  present  chapter. 

6.  Prop.  L— If  f(x)  =  Cnxn  +  CU^"1  +  .  .  .  .  C2x2 
-\-Cxx  -f  Co,  he  a  rational  integral  function  of  x,  then  any 
term  Cr  xr,  that  occurs,  may  le  made  to  contain  the  sum  of  all 
the  terms  that  follow  it,  as  often  as  we  please,  by  taking  x 
large  enough. 

Let  k  be  the  numerical  value  of  the  greatest  coefficient  that 
follows  Cr ,  the  sum  of  all  the  terms,  C'r_i  xr~l  -f  CV_2  xr~2  -f  . . . . 
Cxx  +  (70,  that  follow  Crxr,  cannot  be  greater  than 


k  (x1-1  +  xr~2  +  .  .  .  .   x  +  1) 

x— 1 
Cr(x— l)x*        Cr(x-1) 


xr—l 

that   is,   than   k  ■ r  .     The   quotient   of   Gr  xr   by  this   is 


The  numerator  of  this  expression 
k(x'—l)  k(l—x~r)  l 

may  be  made  as  great  as  we  please,  and  the  denominator  as 

near  to  k  as  we  please,  by  taking  x  large  enough. 

Cor.  1. — By  putting  x  =  - ,  tne  function  becomes 

y-»(Cn  +  cLy  +  j.y  .  dtT1  +  C,y% 

in  which  any  term  nray,  as  above  shown,  be  made  to  contain, 
as  often  as  we  please,  the  sum  of  all  the  terms  that  precede  it, 
by  taking  y  great  enough,  that  is,  by  taking  x  small  enough. 


ti 


G  ALGEBRAICAL    EQUATIONS. 

Cor.  2. — When  x  is  large,  the  term  in  f(x)  containing  the 
highest  power  of  x  is  the  most  important,  and  vice  versa. 

Ex.— If  /(*)  =  3a*       -7a*       -5x*       +  Sx       +  11, 
then  /(10)  =  3  x  104-  7  x  103-  5  x  102  +  8  x  10+  11, 

in  which  the  first  term  greatly  exceeds  the  sum  of  all  the 
remaining  terms  in  value  ;  and  in 

/(rV)   =  3(A)4  -  ?(tV)3  -  5(jV)2  +  8(tV)  +  11, 
the  last  term  exceeds  the  sum  of  all  the  preceding  terms. 

7.  Prop.  II. — To  determine  the  form  that  f(x)  assumes 
when  x  +  y  is  put  for  x,  that  is,  when  x  is  increased  or  dimin- 
ished by  any  quantity  y.     Let 

/ (x)      —  Cn xn        +  Cn-i xn~l       +..C2x2        +  d x+C0;  then 
f(x  +  7j)  =  ClXx  +  2jY+C^x  +  yy-l  +  ..C2(x  +  yy+C1(x  +  tj)  +  C0. 

Expanding  the  different  powers  of  x  +  y  in  this  series,  by 
means  of  the  Binomial  Theorem,  and  arranging  the  result 
according  to  ascending  powers  of  y,  we  obtain, 

f{x-\-y)  =     Cnxn  +  CnUaT-1  +  .  .  .  .  CUx1  +  C,x  +  C0 

+  y  \nCnxn~l  +  (n-l)Cn^x-2 

(  +  (n-2)Cn.2xn-^....Cll 


+  f  Ui(n-l)Cn 


+  {n—1)  ()i-2)Cn-lxn-^  +  ...  .26 


'} 


+  y\n{n-l){n-2)Cnx^ 

£   c    +(^_i)(M_2)(w-3)CU^-4+-.3-2-C'3|- 

+ 

+  f  Ui(n-1). .  .(n-r  +  l)Cnx»-r 

£   <     +  (n _  1) (n _ 2) . .  (n  -  r)  Cn.xx^  +  . .  \r_  Cr  j- 


f(fe«i- 


FUNDAMENTAL   PROPERTIES   OF   EQUATIONS.  7 

Thus  we  see  that  the  development  of  f{x  +  y)  is  a  series 
of  the  form, 

P+  Q9  +  b£  +  *s£+ C„f, 

where  P,  Q,  R,  &c,  represent  the  compound  coefficients  above, 
and  are  all  functions  of  x. 

The  first  of  the  coefficients  P,  is  evidently  the  original 
function  f(x).  The  second  coefficient  Q,  is  derived  from  V 
by  multiplying  each  term  in  P  by  the  exponent  of  x  in  that 
term,  and  then  diminishing  the  exponent  by  unity,  that  is 
(Art.  4),  Q  is  the  first  derived  function  of  /(#),  which  we 
denote  by  f(x).  P,  in  like  manner,  is  the  second  derived 
function  f2(x),  S  is  the  third  derived  function  f(x),  and  so  on, 

ur 
f  (x)  being  the  coefficient  of  — . 

Employing  this  notation,  we  have, 

/(*+»)  =/(s)+/i(%+/2(*)|+  ■  •  :M*)&  ■  •  ■/.(*)£. 

where   —fn  (x)  is  evidently  Cn . 

\n 

By  the  student  acquainted  with  the  Differential  Calculus, 
this  expression  for  f(x  +  y)  will  be  recognized  as  a  particular 
application  of  Taylor's  Theorem. 

8.   Prop.  III. —  If  in  the  function, 
f(x)  =   Cna?  +  CU^1  +  •  •  •  .  C2a*  +  C\x  +  C0, 

x  vary  continuously  from  a  to  b,  then  mil  f(x)  vary  contin- 
uously from  f(a)  to  f(b). 

For  let  c  be  any  value  given  to  x  between  a  and  I,  then 
/(c)  will  be  the  corresponding  value  of  f(x).  Let  c  +  h 
be  another  value  of  jc,  then,  Prop.  II, 

f{e+h)  =  f(C)  +  A(c)h  +/■(«)£+ . .  ./.-.(^  +/.(*)£"• 

Now,  Prop.  I,  Cor.  1,  by  taking  //  small  enough,  the  first 
term  of  the  above  series,  after  /(c),  that  does  not  vanish,  may 


8  ALGEBRAICAL    EQUATIONS. 

be  made  to  contain  the  terms  that  follow  it,  as  often  as  we 
please,  and  this  term  itself  may  be  made  as  small  as  we  please : 
that  is,  f(c-\-h)  may  be  made  to  differ  from  /(c)  by  a  quan- 
tity as  small  as  we  please. 

Hence,  if  in  a  function  f(x),  x  be  supposed  to  pass  in  suc- 
cession through  all  the  values  from  a  to  b,  the  function  will 
also  pass  through  all  the  values  from  f(a)  to  f(b),  seeing  that 
whatever  intermediate  value  f(c)  it  may  have,  we  may  find 
another  as  near  to  it  as  we  please  by  making  a  suitable  change 
in  the  value  of  x.     For  example, 

Let   f(x)    =    x5  —  18z2        +  101s        —  180, 
then      /(l)    =    1—18  +101  —  180  =  -96, 

/(10)  =  103  —  18  x  102  -f  101  x  10  —  180  =      30. 

Here  we  see  that  for  x  =  1,  the  value  of  the  function  is 
—  96 ;  for  x  =  10,  the  value  is  30.  What  the  proposition 
asserts  is  that  the  function  may  be  made  to  pass  through  all 
the  values  between  —  96  and  30,  by  giving  to  x  all  the  values 
between  1  and  10.  It  is  not  asserted  that  the  function  is  al- 
ways increasing  in  this  interval.  In  point  of  fact,  this  particu- 
lar function  will,  on  trial,  be  found  to  be  sometimes  increasing, 
and  sometimes  decreasing,  between  the  limits  —  96  and  30. 

9.  Prop.  IV. — If  in  the  equation, 

f(x)  =  Cnn»  +  CUtf""1  +  .  .  •  .  C2x2  +  Cxx  +  C0  =  0, 

two  numbers,  separately  substituted  for  x,  give  'results  with 
contrary  signs,  one  root,  at  least,  of  the  equation  must  lie 
between  these  numbers. 

Let  a  and  b  denote  the  numbers.  Then,  since,  by  suppo- 
sition, f(a)  and  f(b)  have  contrary  signs,  and,  Prop.  Ill,  as  x 
increases  continuously  from  a  to  b,f(x)  must  pass  continuously 
through  all  the  values  from  f(a)  to  f(b),  one  of  these  values 
must  be  zero,  as  a  series  of  finite  quantities  cannot  pass  from 
one  sign  to  another,  without  passing  through  zero. 

10.  In  the  example  given  in  Art.  8,  wTe  found  /(l)  =  —96, 
f(10)  =  30.  By  the  present  theorem,  there  must  be,  there- 
fore, at  least  one  root  of  the  equation  a?  —  18a;2  -f  101#  —  180 


FUNDAMENTAL   PROPERTIES   OF   EQUATIONS.  9 

=  0,  between  1  and  10 ;  it  may  be  readily  ascertained  thai 
there  are  three.  Again,  though  for  /(l)  we  obtain  —  96,  and 
for  /(6),  the  value  —  G,  we  must  not  conclude  that  there  is 
no  root  of  the  equation  between  1  and  6 ;  there  are  in  fact 
two. 

11.  Prop.  V. —  Every  equation  of  an  odd  degree  has,  at 
least,  one  real  root,  of  a  sign  contrary  to  that  of  the  final 
term. 

In  f(x)  =  Cnx»  +  CUa"-1  +  .  .  .  C2x2  +  C\x  +  C0  =  0, 
suppose  n  to  be  an  odd  number. 

When  we  put  x  =  0,  the  function  reduces  to  the  final 
term  CQ,  and  has  the  sign  of  th^t  term.  By  taking  x  large 
enough,  we  may,  Prop.  I,  make  the  first  term  Cn  xn  greater 
than  the  sum  of  all  the  remaining  terms,  and  the  function 
will  have  the  same  sign  as  x,  which  sign  may  be  taken  contrary 
to  that  of  6q  . 

Since,  then,  /(0)  has  the  sign  of  CQ ,  and  f(a),  by  taking  a 
large  enough,  and  of  sign  contrary  to  that  of  C0 ,  has  the  con- 
trary sign,  there  must,  Prop.  IV,  be  a  real  root  of  f(x)  =  0 
between  0,  and  a,  that  is,  a  root  of  sign  contrary  to  that  of  the 
final  term  C0 . 

Ex.  1.  7x5  +  23a4  —  16z2  +  8x  +  19  =  0, 

must  have,  at  least,  one  negative  root. 

Ex.  2.  x1  +  17.T4  —  105z3  +  27z  —  301  =  0, 
must  have,  at  least,  one  positive  root. 

12.  Prop.  VI. —  Every  equation  of  an  even  degree,  having 
the  final  term  negative,  has,  at  least,  two  real  roots,  one  of 
each 


Let  the  equation  be, 

f(x)  =   CHz*  +  CU&*  +  •  •  •  •  02x2  +  dx  +  C0  =  0. 

When  x  =  0,  the  function  is  negative,  by  hypothesis,  and 
when  x  is  taken  large  enough,  the  function  is  positive,  whether 
x  is  positive  or  negative,  since  xn  is  of  even  degree.  The 
equation  has,  therefore,  at  least  two  roots,  one  between  zero 


10  ALGEBRAICAL    EQUATIONS. 

and  some  positive  number,  the  other  between  zero  and  some 
negative  number. 

Ex.       03*  -  13a?  +  23a*  +  ~'2x«  —  19a?  —  56  =  0.. 

must  have,  at  least,   two   real   roots,   one   positive,  and  one 
negative. 

13.  If  we  could  now  prove  that  every  equation  of  an  even 
degree,  with  its  final  term  positive,  has  a  root,  we  should  have 
established  the  general  proposition  that  every  equation  has  a 
root,  the  basis  upon  which  the  whole  of  the  subsequent  theory 
of  equations  may  be  established.  To  know,  however,  that 
every  equation  has  not  necessarily  a  real  root,  we  do  not 
require  to  go  beyond  the  lowest  of  equations  of  an  even 
degree,  that  is,  quadratics,  which  may  not  have  any  real  root 
when  the  final  term  is  positive.  The  question,  accordingly, 
narrows  down  to  this.  Have  all  equations  of  even  degree,  with 
the  final  term  positive,  roots,  if  not  real,  imaginary,  similar  to 
those  obtained  for  equations  of  the  second  degree  ?  The 
investigation  of  this  problem  will  be  given  in  Chapter  II. 
Before  proceeding  to  that  part  of  the  subject,  we  shall  prove  a 
few  propositions  which  are  independent  of  the  results  of  that 
chapter. 

14.  Prop.  VII. — If  a  function  f(x),  and  the  successive 
quotients  arising  from  the  division,  be  divided  by  x  —  a,  the 

remainders,  in  succession,  ivill  be   f(a),  f(a),  775-/2  (#)»  .... 

Let  the  function,  which  we  shall  suppose  of  the  nth  degree, 
be  divided  by  x  —  a.  As  the  divisor  contains  only  the  first 
power  of  x9  we  shall  evidently  arrive  at  a  remainder  independ- 
ent of  x,  and  the  quotient,  like  the  dividend,  can  contain  only 
positive  integral  powers  of  x.  Denoting  the  quotient  by  Q, 
and  the  remainder  by  R,  we  have 

f(x)  =  {x-a)Q  +  R  [1]. 

If  Q,  which  is  a  function  one  degree  lower  than  f(x),  be  in 


FUNDAMENTAL   PROPERTIES   OF   EQUATIONS.  11 

its  turn  divided  by  x  —  a,  then,  denoting  the  quotient  by  Qx , 
and  the  remainder  by  Rv ,  we  have 

Q  =   Q^x-a)  +  ft.  [2]. 

Substituting  this  value  of  Q  in  [1],  we  obtain 

f{x)  =  Q,{x-af  +  ft(*-a)  +  i?. 

By  continuing  in  this  manner  to  divide  each  successive 
quotient  by  x  —  a,  and  substituting  its  equivalent  for  the 
symbol  for  that  quotient  in  the  preceding  identity,  we  evi- 
dently, after  n  divisions,  shall  obtain  as  quotient  the  simple 
factor  Cn,  the  coefficient  of  the  first  term  of  f(x),  and  have 

f(x)  =  Cn(x-a)n+Rn-1(x-a)n-l+  .  .B2(x-a)*  +  Rl(x-a)  +  R. 

In  this  result,  putting  y  =  x  —  a,  wre  have 

/(a+y)  =  ay+*-ir"l+  •  •  •B2f+E1y+E.    [3] 

Now,  by  Art.  7,  we  have,  writing  the  terms  in  reverse  order, 

/(a+jr)  =  C'„y"+]^r1/»-.(«)r-1+ . .  .|-/2(«)2/2+/.(% 

+/(«)      w- 

As  the  coefficients  of  equal  powers  of  y  in  these  identities 
must  be  equal,  we  find 

*=/(«),  ft=/i(«),  A=j|-/i(fl)f-.-..«^i=]^|A-*W; 

that  is,  tf*0  successive  remainders  arising  from  the  division 
ly  x  —  a  of  f{x),  and  the  successive  quotients,  will  be,  first, 
f:(a),  the  value  of  the  function  when  a  is  put  for  x  ;  second, 
f(a),  the  value  of  the  first  derived  function  of  fix),  when 
a  is  put  for  x,  and  so  on. 

15.  Cor.  1. — A  function,  f(x),  is  exactly  divisible  by 
x  —  a,  if  a  is  a  root  of  fix)  =  0;  for  then  f(a)  =  0. 

16.  Cor.  2. —  Conversely,  if  f(x)  is  exactly  divisible  by 
x  —  a,   then  a   is  a  root  of  the   equation  f(x)  =  0;    i.  e., 

/w  =  o. 


12  ALGEBRAICAL  EQUATIONS. 

17.  In  the  next  proposition  will  be  given  a  convenient 
process  for  finding  the  successive  quotients  and  remainders 
without  actual  division.  Yet  as  this  proposition  is  of  con- 
siderable practical  importance,  being  the  basis  of  Horner's 
Method,  the  student  would  do  well  to  work  one  or  two  exam- 
ples by  dividing  in  the  usual  way,  and  compare  the  successive 
remainders  with  the  results  obtained  by  actual  substitution  in 
the  primitive  and  derived  functions. 

Thus  suppose  we  take  the  function, 

f(x)   —     3a5  —  13a4  —  21a3  +  10a2  +  91a  +  425. 
then,       fx{x)  =  15a4  —  52a3  —  63a?  +  20a  +  91, 

i-/2(a)   =  30a3  -  78a*  -  63a  +  10, 
12 

jg-/a(aO  =  30a2  —  52a  -  21, 
ji-/4(a)   =  15a  -  13, 

g/i(*)  =     3. 

If  now  we  divide  f(x)  by  a  —  3,  we  obtain  as  remainder 
—  103,  which  is  the  value  of  /(3),  i.  e.,  of  the  function  when 
3  is  put  for  a.  If  we  divide  the  first  quotient  by  a  —  3,  we 
obtain  as  remainder  —  605,  the  value  of  /i(3),  and  so  on  with 
the  other  remainders. 

EXERCISES. 

18.  Prove  by  division  that 

3  is  a  root  of    a3  —    W  —      32a  +  132  =  0. 

5  "      "    of  4a5  —  11a3  +      20a  -      11225  =  0. 

6  «      «     of  3a3  —  25a3  —  1358a2  —  1625328  =  0. 

19.  Prop.  VIII. —  To  find  the  quotient  and  remainder 
wJien  /(a)  =  Cnxn  +  CUs*"1  +  .  .  .  .  C.x2  +  Cxx  +  C09 
is  divided  by  x  —  a. 

Supposing  the  division  to  be  performed  till  we  arrive  at  a 
remainder  independent  of  a,  and  denoting  the  quotient  by  Q 
and  the  remainder  by  /?,  we  have 


FUNDAMENTAL   PROPERTIES    OF   EQUATIONS.  13 

f(x)  or  CU"  +  CLiaJ""1  +  •  •  .  C2x2  +  Cxx  +  ft 

=  (a?-a)e  +  &         [!]• 

By  Art.  14,  we  know  that  i?  =  f(a),  that  is, 

OX  +  GLicrH  . . .  -ftrf  +  ftfl  +  ft  =  5.        [2], 

Subtracting  [2]  from  [1] ,  we  have 
£,(#•— d*)  +  CU(af-*— a""1)  +  . . . .  C'2(z2-«2)  +  C^— a) 

The  binomials  xn—a'\  xn~l—an~l,  &c,  are  all  divisible  by 
ic  —  «,  as  proved  in  elementary  algebra.  Effecting  the  di- 
vision, we  obtain, 

Cn(xn-X  +  ax'1-1  +  axn~2  +  .  .  .  .  an~2x  -f-  a""1) 
+  Cn_x(xn-'2  +  oaf*  +  .  .  .  .  a"-2) 

+ 

+  C2(aJH-fl) 
+  C\     =      Q. 

Arranging  this  result  according  to  powers  of  x,  we  have 

Gn  xn~l  +  ( Cn  a  +  CU)  Z""2  +  ( QL  «2  +  CL,  a  +  CU)  af* 

+  ( Cn  a?  +  ft-i  a2  +  0,-2 «  +  C;_3)  z-4  +  . . . . 

+  ((7n«"-1+(7„_1^-2-f....(73«2+62«+C/1)  =  0. 

Thus  we  find  that,  in  the  quotient  of  f(x)  by  x  —  a,  the 
coefficient  of  the  first  term  is  Cn9  and  the  remaining  coefficients 
are  formed  in  succession  by  multiplying  each  term  in  the 
preceding  coefficient  by  a,  and  then  adding  in  the  correspond- 
ing coefficient  of  f(x). 

20.   Upon  examining  the  final  coefficient  of  Q,  namely, 

CnCT1  +  Cn.xan'2  +  .  .  .  .  Cza2  +  C,a  +  Cl9 

we  see  that  by  applying  the  process  to  this  coefficient  also, 
and  adding  in  C0 ,  we  obtain 

Cltan  +  CUflT1  +  .  •  •  .  C,a?  +  C,d2  +  C\a  +  C0, 

which,  as  before  shown,  is  the  remainder,  f(a) . 


14  ALGEBRAICAL    EQUATIONS. 

21.  By  this  process  the  operation  of  division,  so  tedious  by 
the  usual  method,  is  reduced  to  an  operation  both  simple  and 
compact.  In  practice  we  proceed  as  follows :  We  arrange  the 
coefficients  of  the  function  to  be  divided  in  a  horizontal  row, 
supplying  by  zeros  the  coefficients  of  any  terms  that  may  be 
absent.  Supposing  x  —  a  to  be  the  divisor,  we  multiply  the 
first  coefficient  of  the  function  by  «,  and  add  in  the  second ; 
we  multiply  this  result  again  by  a,  and  add  in  the  third 
coefficient,  and  so  on  till  we  have  added  in  the  last  coefficient. 
The  final  result  will  be  the  remainder,  which,  as  we  have  seen, 
is  also  the  value  of  the  function  when  x  =  a.  The  preceding 
results  will  be  the  coefficients  of  the  quotient,  in  order,  with 
their  proper  signs.  The  first  coefficient  of  the  function,  which 
is  also  that  of  the  quotient,  we  supply  from  the  left  hand  of 
the  upper  row  of  coefficients. 

Ex.  1. —  Required  the  quotient  and  remainder  arising  from 
the  division  of 

xi  _  5.^6  +  19^5  _  32^4  _  63a?  +  og^a  +  73^  +  42 

by  the  binomial  x  —  3. 

We  proceed  as  follows : 

1  _  5  +  19  _  32  —  63  +    98  +  73  +  42    \J>_ 
3  _    6  +  30  +  21  —  126  —  84  —  33 
_  2  +  13  +    7  —  42  -    28  —  11  +    9. 
Here  the  quotient  is 

a«  _  2x5  +  13a4  +  7^  _  42^  —  28x  —  11, 

and  the  remainder  is  9,  which  is  also  the  value  of  the  given 
function  when  x  =  3. 

Ex.  2. —  Required  the  quotient  and  remainder  arising  from 
the  division  of  the  function 

5^8  _  466:C5  _  63734  _  760^3  _  142a?  _  370, 

by  the  binomial  x  —  5.  , 

5  +    0  +      0  -  466  -  637  —  760  -  142  +    0  -  370   |  5 

25  +  125  +  625  +  795  +  790  +  150  +  40  +  200 

25  +  125  +  159  +  158  +    30  +      8  +  40  —  jlTO 


FUNDAMENTAL    PROPERTIES   OF   EQUATIONS.  15 

Here  the  quotient  is 

5a7  +  25a6  +  125a5  +  159a*  +  158a;3  +  30a;2  +  8x  +  40, 
and  the  remainder  is  —  170. 

22.  By  applying  the  same  process  to  the  first  quotient,  we 
obtain  a  second  quotient,  and  a  second  remainder,  which,  as 
shown  in  Art.  14,  is  the  value  of  /^a)  for  x  =  a ;  from  the 
second  quotient,  in  like  manner,  we  obtain  a  third  quotient 
and  a  third  remainder,  which  is  the  value  of  %f>{x)  for  x  =  a. 
and  so  on.  This  will  be  more  fully  illustrated  in  Art.  69. 
As  the  process  thus  has  its  most  useful  application  in  finding 
the  value  of  a  given  function  for  an  assigned  value  of  x,  we 
shall  give  a  few 

EXERCISES. 

Evaluate  the  following  functions  : 

1.  x5  —  7a4  +  43a3  —  78a;2  +  103a;  —  260,   for  x  =  3. 

2.  8a;6  +  13a5  —  99a4  —  247a;3  —  387a-2  +  107a;  +  638,  for 
x  =  4. 

3.  5a;7  —  646a4  +  496.r2  —  160a;  -  121,   for   x  —  5. 

4.  a8  +  73a;3  +  505a-2  +  1059a  +  1875,   for  x  =  —  2. 

5.  a?  —  73a;6  +  54a4  —  93a2  +  101,   for  a  =  11. 


16  ALGEBRAICAL    EQUATIONS. 


CHAPTEK       II. 
IMAGINARY    EXPRESSIONS  —  CAUCHY'S    THEOREM. 

23.  As  before  adverted  to  (13),  we  need  not  extend  our 
inquiries  beyond  equations  of  the  second  degree  to  find  that  it 
cannot  be  said,  generally,  that  every  equation  has  a  real  root. 
We  find,  in  fact,  that  when  certain  relations  exist  among  the 
coefficients  of  a  quadratic,  we  obtain  as  roots,  not  real  quan- 
tities, but  expressions  of  the  form  a  -f  bV—  1,  in  which  b 
is  not  zero.  Though  these,  when  substituted  for  x  in  the 
given  equation,  cause  it  to  vanish,  and  are  thus  roots  by  defi- 
nition (2),  yet  as  they  indicate  operations  to  which  no  arith- 
metical meaning  can  be  attached,  they  have  received  the  name 
of  imaginary,  or,  perhaps  more  appropriately,  impossible  roots. 

We  propose  in  the  present  chapter  to  show  that  every  equa- 
tion has  a  root  of  the  form  a  +  b  V—  1,  in  which  a  and  b 
are  real  quantities,  positive,  negative,  or  zero.  When  b  is  zero, 
the  root  is  real,  in  all  other  cases  imaginary. 

24.  Since  these  imaginary  expressions  are  of  constant  and 
unavoidable  occurrence  in  the  theory  of  equations,  it  may  be 
found  convenient  to  give  a  summary  of  the  chief  results  arising 
from  the  conventions  adopted  in  regard  to  them. 

The  first  convention  is  that  we  regard  ±  V—  «2  as  equiva- 
lent to  ±  a  a/—  1,  since  thus  we  introduce  but  one  imaginary 
symbol  into  our  investigations,  namely  V—  1.  The  whole 
expression  a  -+-  bV —  1,  consisting  of  a  real  part  a,  and 
another  real  part  b  affected  by  the  sign  of  an  impossible 
operation,  we  consider  as  imaginary  on  account  of  the  presence 
of  this  latter  part.  We  also  regard  such  terms  as  b  V—  1  as 
subject  to  the  ordinary  rules  that  govern  algebraical  trans- 
formations. 


IMAGINARY    EXPRESSIONS.  17 

25,  From  these  conventions  Ave  obtain  the  following 
results : 

I.  The  sum  or  difference  of  expressions  of  the  form 
A  +  B  V—  1,  have  the  same  form  ;  thus, 

(a  ±  bV^)  ±  {a1  +  &V=T)  = 
(a±a!)  ±  (b  ±6')Virl, 

which  is  of  the  form  A  +  BV—  1. 

II.  The  product  of  expressions  of  the  form  A  +  B  V—  1  is 
of  the  same  form  ;  thus, 

[a±bV~-L)  x  (a'±b'V^l)  =  (aa'-bb')  ±  (a'b  +  ab')\/~-L. 

III.  The  quotient  of  one  expression  of  the  form  A  +  BV— 1> 
by  another  of  that  form,  is  still  of  the  same  form  ;  thus, 

a  ±bV^l     _  (a  ±b  V^l)  (a'  T  ft'V^l)   _ 
a'  ±  j'v^TI  " "   {a'  ±  b'V^l)  (a'  =F  bW^^l)  ' 

{ad  ±  bV)  ±  (a'b  -  ab')V^l 
a'*  +   V* 

which  is  still  of  the  given  form. 

26.  From  the  result  obtained  in  II,  we  infer  that  any 
positive  integral  power  of  a  +  b  V  —  1  is  of  the  same  form ; 
and  may  thence  infer  that  if  a  -f-  I  V—  1  be  substituted  for  x 
in  a  rational  integral  function  of  x,  whether  the  coefficients 
be  all  real,  or  any  of  them  imaginary,  we  shall  obtain  an  ex- 
pression of  the  form  P  -f  Q*\/—l. 

2H,  Two  imaginary  expressions  a  +  bV  —  1,  and  a— bV— 1» 
which  differ  only  in  the  sign  of  the  imaginary  part,  are  said  to 
be  conjugate. 

Hence  the  sum  and  the  product  of  two  conjugate  imaginary 
expressions  are  real.  This  we  see  exemplified  in  the  imagi- 
nary roots,  a  +  bV—  1,  and  a  —  bV—  1,  of  a  quadratic, 
their  sum  2a,  with  changed  sign,  being  the  coefficient  of  x, 
and  their  product,  a2  -f-  J2,  the  final  term. 


18  ALGEBRAICAL  EQUATIONS. 

28.  The  positive  value  of  the  square  root  of  this  product, 
that  is,  Va2  +  b2,  is  called  the  modulus  of  each  of  the  ex- 
pressions a  +  b V— 1,  and  a  —  b  \  —  1.  The  modulus  of  a 
real  quantity  is,  by  the  above  definition,  the  positive  value  of 
the  quantity  itself. 

When,  for  any  purpose,  we  desire  to  compare  imaginary  ex- 
pressions with  each  other  or  with  real  quantities  in  regard  to 
magnitude,  this  can  be  done  only  by  comparing  the  moduli 
of  the  expressions. 

29.  In  order  that  an  imaginary  expression  be  zero,  it  is 
necessary  and  sufficient  that  its  modulus  be  zero  :  for  in  order 
that  Va2  +  I2  may  be  zero,  we  must  have  a  =  0,  and  b  =  0. 

30.  The  product  of  two  quantities  has  for  modulus  the 
product  of  their  moduli ;  thus, 

(a  +  b  V^l)  (a  +  I V^l)  = 
(act'  -  bb')  +  {aV  +  a'fyV^l, 
and  the  modulus  of  this  is, 


V  i  {act  —  bb')2 .+  (aV  +  a'b)2\    =   vV  +  #2)  K2  +  ^'2), 
which  is  the  product  of  the  moduli  of  the  original  expressions. 
In  a  similar  manner  it  may  be  shown  that  the  modulus  of 
the  quotient  of  two  quantities,  is  the  quotient  of  the  modulus 
of  the  dividend  by  the  modulus  of  the  divisor. 

31.  If  we  raise  V —  1  to  the  second  power,  we  obtain  —1 : 
this  multiplied  by  V—  1,  gives  the  third  power  —  V—  1 ; 
This  again  multiplied  by  V—  1?  gives  the  fourth  power  4-  1. 
Upon  proceeding  to  higher  powers,  we  obtain  a  recurrence 
of  the  preceding  results,  V—  h  —  1,  —  V—  1?  +  1-  Even- 
whole  number  must  be  of  one  of  the  four  forms,  4r,  4r  -f  1. 
4r  +  2,  4r  +  3,  since,  when  divided  by  4,  it  must  leave  as 
remainder,  0,  1,  2,  or  3.  "We  have,  therefore,  all  possible 
integral  powers  of  V —  1  in  the  four  forms, 

=  1,    (V^i)ir+1  =  V'-i,    (V^i)4+2  =  -1, 


IMAGINARY    EXPRESSIONS.  19 

32.   In  elementary  algebra  is  given  the  formula  for  the 
square  of  an  expression  of  the  form  a  -j-  b  V —  1, 


\/a  +  bV-l  =  \i( Vtf  +  P  +  a)  f*+  \  UVaZ  +  P-a)  \$V-1. 

From  this  we  may  obtain  the  square  roots  of  ±  V—  1,  by 
putting  a  —  0,  b  =  ±1.     Thus  we  obtain 

33.  Before  proceeding  to  the  general  proposition  that 
every  equation  has  a  root,  it  will  be  found  convenient  to 
discuss  certain  equations  of  very  simple  form. 

Prop.  I. —  Each  of  the  equations, 

xn  —    ±1,         xn  =    ±  a/^1, 

has  a  root,  real  or  imaginary. 

I.  x11  —  +  1.  This  equation  evidently  has  a  real  root, 
whether  n  be  an  odd  or  an  even  number,  since  x  =  1  satis- 
fies it  in  either  case. 

II.  x11  =  —  1.  When  n  is  an  odd  number,  this  equation 
is  obviously  satisfied  by  x  =  —  1. 

When  n  is  an  even  number,  it  must  be  of  the  form  2m ; 
that  is,  we  may  put  the  equation  under  the  form  y2m  =  —  1. 
Taking  the  square  root  of  both  members,  we  obtain 
if1  =  ±  V—  1,  an  equation  of  the  forms  we  are  about  to 
consider. 

III.  xn  =  +  V—  1.  When  n  is  an  odd  number,  it  must 
be  of  the  form  4r  + 1,  or  4r  -f-  3.  In  the  first  case,  x  =  +V—1 
is  a  root,  since  (31),  (+  V— 1)4'+1  =  +  V—  1 ;  in  the 
second  case,   x  =  —  V—  1    is   a   root,  since    (—  V—  l)4r+3 

When  n  is  an  even  number,  it  must  be  either  some  power 
of  2,  or  some  power  of  2  multiplied  by  an  odd  number. 
In  the  first  case  suppose  n  =  2"1,  that  is,  put  the  equation 
under   the   form    y°-m  =  +  V—  1.      We   can   now   obtain   a 


20  ALGEBRAICAL  EQUATIONS. 

value  of  y  by  m  successive  extractions  of  the  square  root, 
which  (32)  will  yield  a  result  of  the  form  a  +  b  V  —  1.  In 
the  second  case  suppose  n  =  mp,  where  m  is  an  odd  number, 
and  p  some  power  of  2.  By  putting  y  =  xp,  the  equation 
may  be  put  under  the  form  ym  =  -+-  V—  1,  a  root  of  which, 
as  shown  above,  must  be  y  —  ±  V—  1,  or  #*  =  ±  V—  1, 
the  upper  sign  being  taken  when  m  is  of  the  form  4r  +  1, 
the  lower  when  it  is  of  the  form  4-r  +  3.  We  can  now  find, 
as  above,  a  root  of  x*  —  ±  V  —  1,  p  being  a  power  of  2. 

IV.  xn  =  —  V—  1.  Proceeding  as  in  III,  we  find,  when 
n  is  odd,  x  =  —  V—  1?  or  +  V—  1,  according  as  n  is  of 
the  form  4r  +  1,  or  4r  +  3.  When  n  is  an  even  number, 
we  may  put  it  equal  to  mp,  m  being  an  odd  number,  and  p 
some  power  of  2,  and  so  find  a  root  of  the  form  a  -f  bV— 1. 

Thus,  in  every  case,  an  equation  of  any  of  the  given  forms 
has  a  root  coming  under  the  general  form  a  +  b  V—  1. 

34.  Prop.  II.  —  Every  rational  integral  equation  has  a 
root  of  the  general  form    a  +  b  V—  1. 

Let     /(a)  ==  ftaf  +  CUaf"1  +  . . . .  <72z2  +  Cxx  +  C0, 

in  which  the  coefficients  C;i,   Cn_i, Cl9  C0,  maybe  either 

real  or  imaginary. 

If,  in  this  function,  we  substitute  a  +  b  V—  1  for  jc, 
«  and  b  being  real,  we  shall  obtain  (26)  a  result  of  the  form 
P  +  QV—  1,  i5  and  §  being  real.  Now,  in  order  that 
a  +  b  V—  1  may  be  a  root  of  /(z)  =  0,  this  result  must  be 
zero,  that  is,  P  and  Q  must  be  simultaneously  zero,  and  there- 
fore the  modulus  \/P2  +  Q2  also  zero.  It  is  required  to  show 
that  some  value  of  a  +  b  V—  1  must  exist,  for  which 
VP2  +  Q2  becomes  zero.  For,  if  it  could  not  become  zero, 
there  would  be  some  value  below  which  it  could  not  be  dimin- 
ished. But  it  will  be  proved  that  whatever  value  of  */P2+  Q2, 
different  from  zero,  can  be  obtained,  a  value  still  smaller  can 
be  obtained  by  making  a  suitable  change  in  the  expression 
that  is  substituted  for  x  in  the  function.  VP2  +  Q2,  there- 
fore, must  be  capable  of  becoming  zero  for  some  value  of 


IMAGINARY    EXPRESSIONS.  21 

a  +  b  V—  1,  that  is,  the  function  must  become  zero  for  some 
value  of  x. 

Let  us  suppose  that  for  some  assigned  value  of  re,  as 
x  =  a  +  bV—  1,  we  obtain 

/(*)  =  p+  ov^i.  [i] 

in  which  P  and  §  are  not  both  zero. 

If  we  change  a  +  bV—  1  to  a  +  bV—  1  +  &,  that  is, 
x  to  jc  +  7j,  we  have,  Art.  7, 

/(*+*)  =/(i)+/i(*)*+/^)^|-+  •  •  •  /.(*)*£■  m. 

In  [1]  we  have,  for  z  =  «  +  b  a/^1,  f(x)  =  P  +  Q^~—L ; 
for  this  value  of  a;  some  of  the  derived  functions  /i(#),  ./K^), 
&c.,  in  [2]  may  possibly  become  zero,  but  they  cannot  all  be- 
come zero,  fn(x),  at  least,  which  (7)  does  not  contain  x,  must 
remain.  Suppose  hm  to  be  the  lowest  power  of  li  whose  co- 
efficient does  not  vanish  in  [2].  Having  denoted  the  value 
of  f(x)  by  P  -f  CV~1,  so  we  may  put  P'-f  #V— 1  for 
the  value  of  f(x  +  h),  and  R  +  #a/—  1,  in  which  P  and  $ 
are  not  both  zero,  for  the  value  of  that  derived  function  which 
is  the  coefficient  of  hm. 

Substituting  these  values  in  [2],  we  have, 

F  +  Q'V-1  =  {P+Q  V~-l)  +  (R  +  S  V-I)  *» 

+  terms  in  Aw+1,  hm+2,  &c.     [3]. 

As  we  may  assign  any  value  we  please  to  h,  we  may  replace 
it  by  Jet,  where  Jc  is  a  real  positive  quantity,  and  t  is  such 
that  (33)  tm  may  be  +  1,  or  —  1,  as  we  choose.  Hence  we 
can  make  F+  Q'V^l  =  (P+QV~^l  ±  (R  +  SV^l)*?" 
-f  terms  in  higher  powers  of  7c. 

In  such  an  equation  the  sum  of  the  real  terms  in  one  mem- 
ber must  be  equal  to  the  sum  of  the  real  terms  in  the  other, 
and  the  imaginary  terms  in  the  one  to  the  imaginary  terms  in 
the  other : 

.%     F  —  P  ±  RJcm  +  terms  in  hm+\  Jcm+2,  &c. 
Q'  —  Q  ±  Sic"1  +  terms  in  Jcm+\  Jcm+2,  &c. 

...     p'2+  Q'2  _  (p2  +  Q2)  ±  2  (PR  +  QS)^  +  other 

terms  in  km+1,  &c.    [4]. 


22  ALGEBRAICAL    EQUATIONS. 

Now  k  may  be  taken  so  small  that  (Art.  6,  Cor.  I)  the  sum 
of  all  the  terms  after  P2  4  Q2  will  have  the  same  sign  as 
2  (PR  +  QS)  km,  provided  PR  +  QS  be  not  zero. 

First,  supposing  PR  4  QS  is  not  zero,  then  the  sign  of 
(P'2+0'2)_(p2+02)  is  the  same  as  that  of  ±  %(PR+QS)km, 
when  k  is  taken  small  enough,  and  this  sign  we  can  always 
make  negative  by  taking  t  such  that  t m  =  —  1,  when 
PR  +  QS  is  positive,  and  vice  versa.  We  can  thus  always 
make  P'2  +  Q'2  less  than  P2  +  §2. 

If  PR  +  ()#  be  zero,  then  we  choose  t  so' as  to  make  nega- 
tive the  first  term  after  2(PR-\-QS)lm  that  does  not  vanish 
in  [4],  and  as  before  we  obtain  (P'2  +  Q'2)  —  (P2  +  Q2)  =  a 
negative  quantity,  that  is,  we  have  VP'2+Q'2  <  VP2+Q2. 


Thus,  whatever  be  the  value  of  the  modulus  yP2  4  Q2,  a 
modulus  VP'2  4-  §'2  of  smaller  value  may  be  obtained  by 
making  a  suitable  change  in  the  expression  that  is  substituted 
for  x  in  f(x) ;  there  must,  therefore,  be  some  expression  which, 
substituted  for  x,  will  make  the  modulus  zero. 

35.  Prop.  III.  —  The  values  of  a  and  b  in  the  expression 
a  4  bV—1,  which,  when  put  for  x  in  f(x),  causes  it  to 
vanish,  are  finite. 

We  may  write  f(x)  as  follows  : 

Substituting  a  +  bV—l  for  x  in  the  second  member,  it 
becomes, 


a 


(a  +  bV-iyh  + Cn-\ + 


Cn(a  +  bV-l)       Cn(a  +  bV-l)2 

4.... a=,i. 

Cn(a  +  bV-l)n) 
Selecting  any  term  from  the  series  within  the  brackets,  the 
third,  for  example,  we  have, 

0,-2  _.   Cn_2(a-b\r^i)2 

C„(a  +  bV~-^)2  "  Cn(a2  +  b2)2  *? 

-  c»-2(<*-P)  _  ZC^abV^l  _  A   ,    -  /-- 

~    CH  [a2  4  b2)2         Cn  {a2  4  b2)2     ~  *  +  ^ V  ~ ij  sa* ' 


IMAGINARY    EXPRESSIONS.  23 

Iii  this  it  is  obvious  that  A  and  B  dimmish  without  limit, 
as  a  and  b  increase  without  limit.  Denoting  the  value  of 
f(x),  for  x  =  a  +  b  V^l,  by  P  +  Q  V—  1 ;  we  have 

P+QV-^l   =   Cn(a  +  b^-\y\l+A'+ BW~-i\> 

in  which  A'  and  i?'  diminish  without  limit,  as  a  and  £ 
increase  without  limit.  In  the  same  way  we  have,  for 
x  =  a  —  b\/—  If 

p  _  QV^l  =  C^a-bV^Y  \  1  +  A'-  B'V^l  \  ; 

.-.  p2  +  G2       =  c*(*  +  wy    {  (i  +  A'f  +  B'*     \. 

This  result  increases  without  limit,  when  a  and  b  increase 
without  limit ;  for  then  the  factor  (a2  -f  b2)n  increases  without 
limit,  and  the  factor  \  (1  +  vi')2  +  B'2\  tends  towards  unity, 
as  A'  and  B'  decrease  without  limit.  Thus  the  modulus 
VP2  -f-  Q2,  cannot  become  zero  when  a  or  b,  or  both  of 
them,  are  indefinitely  great. 

By  the  present  proposition,  therefore,  in  conjunction  with 
that  of  Art.  34,  wre  find  that  every  rational  integral  equation, 
whether  its  coefficients  be  real  or  imaginary,  has  a  root  coming 
under  the  general  form  a  -+-  b  V —  1,  where  a  and  b  are  finite 
quantities,  either,  or  both,  positive,  negative,  or  zero. 

An  irrational  equation,  as,  for  example, 

V%  —  4  —  Vx  —  1  —  1   =  0, 

may  be  incapable  of  being  satisfied  by  any  quantity,  real  or 
imaginary,  if  we  regard  the  signs  of  the  roots  indicated  as 
being  controlled  by  the  plus  and  minus  signs  prefixed.  The 
above  equation,  when  rationalized  and  solved  in  the  usual 
manner,  gives  x  =  5,  a  value  that  docs  not  satisfy  the  pro- 
posed equation,  but  does  satisfy  the  equation 

-  y/x  -  4  +  Vx^l  -1   =  0. 


24  ALGEBRAICAL  EQUATIONS. 


CHAPTEK    III. 

GENERAL    PROPERTIES    OF    EQUATIONS,    DEPENDENT    ON   THE 
PRINCIPLE   THAT   EVERY   EQUATION    HAS   A   ROOT. 

36.  Prop.  I.  —  Every  equation  has  as  many  roots  as  it 
has  dimensions,  and  no  more. 

Let  f(x)  =  Cn  a?  +  Cn.x  or1  +  . .  .  C2x2  +  d  x  +  C0  =  0,     [1] 

be  an  equation  of  the  nth  degree,  it  has  n  roots,  and  no  more. 
By  Art.  34,  f(x)  =  0  has  a  root  ai9  real  or  imaginary,  and  is 
therefore  (15)  divisible  by  x  —  ax .     Hence  we  may  put, 

f(x)  =  (x-ai)fa(x),  [2] 

where  0,  (x)  is  a  rational  integral  function  of  the  (n  —  l)th 
degree.  If  fa  (x),  again,  be  equated  to  zero,  it  must  also  have 
some  root  a2,  and  is,  therefore,  divisible  by  x  —  a2,  so  that  we 
have  fa  (x)  =  x  —  x2)  fa(x),  where  fa  (#)  is  a  function  of  the 
(n  —  2)th  degree.  Substituting  this  value  of  fa  (x)  in  [2],  we 
have, 

fix)  =  {x-ax){x-a2)fa{x).  [3]. 

In  like  manner,  fa{x),  equated  to  zero,  has  some  root  «3, 
and  has,  therefore,  a  binomial  divisor  (x  —  a3),  divided  by 
which  it  will  yield  as  quotient  a  function  of  the  (n  —  3)rd  de- 
gree, which  we  may  denote  by  fa  (x) ;  we  have,  therefore, 

f(x)  =  {x-ai)(x-a2)(;x-a,)fa{x).  [4]. 

It  is  evident  that  a  continuance  of  the  process  will  lead 
at  last  to  the  simple  factor  Cn ,  and  that  there  must  be  n  bi- 
nomial factors  (x—ai),  (x—a2),  .  .  .  (x— an) ;  therefore 

f(x)  =  Cn{x—al){x^a.2){x—az)  .  .  .  (x— an)  —  0. 

Hence  the  equation  f(x)  =  0  has  n  roots,  since  the  func- 
tion vanishes  when  any  one  of  the  quantities  ax ,  «2 ,  az ,  .  .  . 
an,  is  put  for  x. 


GENERAL   PROPERTIES   OF   EQUATIONS.  25 

Nor  can  the  equation  have  more  than  n  roots;  for  when  we 
put  for  x  a  quantity  b,  which  is  not  one  of  the  quantities 
tf i ,  # 2 1  <hi  •  •  •  an  9  we  have 

fib)  =   Cn(b-<h)  (b-a2)  {b-a.)  .  .  .  (b-an), 

the  second  member  of  which  cannot  be  zero,  since  none  of  th" 
factors  is  zero. 

37.  Observe  that  it  is  not  asserted  that  the  n  roots  al9  a2, 
a3,  .  .  .  aH,  of  an  equation  of  the  nth  degree  are  necessarily 
different ;  some,  or  all  of  them,  may  be  equal.  What  the  rea- 
soning shows  is  that  the  first  member  of  an  equation  of  the 
nth  degree  may  be  resolved  into  the  product  of  n  binomial 
factors,  each  of  which,  equated  to  zero,  will  furnish  a  root. 
It  is  found  convenient  to  say  that  the  equation  has  as  many 
roots  as  dimensions,  without  regard  to  the  relative  magnitudes 
of  the  roots. 

38.  Cor.  1.  —  From  this  proposition  it  is  evident  that,  when 
a  root  ai  of  the  equation  f(x)  =  0  is  known,  we  can  obtain 
an  equation  containing  all  the  remaining  roots,  by  dividing 
f(x)  by  x  —  al}  and  equating  the  quotient  to  zero,  which 
latter  equation  is  of  one  degree  lower  than  f(x)  =  0.  When 
two  roots,  ai  and  a2i  are  known,  we  can,  by  dividing  f{x) 
successively  by  (x — a^)  and  (x — a2),  or  at  once  by  their  product, 
obtain  a  function  of  x,  two  degrees  lower  than  f(x),  which 
quotient,  equated  to  zero,  will  contain  all  the  remaining  roots ; 
and  similarly  for  any  number  of  roots. 

Ex.      xs  —  18a2  +  101a;  —  180  =  0    has  a  root  x  =  4. 

Dividing  by  x  —  4,  we  obtain  x2  —  14a;  -f  45,  which  equated 
to  zero  gives  two  other  roots,  5,  and  9. 

39.  Cor.  2. — If  a  rational  integral  function  of  x  of  the 
7bth  degree  can  become  zero  for  more  than  n  values  of  x,  then 
every  coefficient  in  the  function  must  be  equal  to  zero,  and 
the  function  must  be  zero  for  any  value  of  x. 

For,  otherwise,  the  function  equated  to  zero  would  have 
more  than  n  roots;   which,  by  the  present  proposition,  is 
2 


2b'  ALGEBRAICAL   EQUATIONS. 

impossible,  unless  the  coefficients  be  each  equal  to  zero  ;  and 
if  these  coefficients  be  each  zero,  the  function  is  evidently 
zero  for  every  value  of  x. 

40.  Cor.   3.— If  f(x)    be    of   the    nth    degree,    it    has 

n(n— l)...(n— r-f-1)  e  „  , ,       ,.  , 

—5 '- 5 factors  of  the  rth  degree. 

\r 

For  it  has  been  shown  that  f(x)  has  n  factors  of  the  first 
degree.  The  product  of  every  two  of  these  simple  factors  is  a 
factor  of  the  second  degree ;  so  there  must  be  as  many  factors 
of  the  second  degree  as  there  are  possible  combinations  of  n 

things  taken  two  at  a  time,  that  is,     ^ ,   — - .    In  like  manner, 

Lf 
with  the  n  simple  factors  may  be  formed  as  many  factors  of 
rth  degree  as  there  are  possible  combinations  of  n  things  taken 

. ,    ,   .      j.,  x     ,     .,      „n{n  —  l) (n  —  /-  +  1) 

r  at  a  time,  that  is,   f(x)  admits  of  — — rT-^ - 

factors  of  the  rth  degree.  These  factors,  of  any  degree,  will 
not,  of  course,  be  all  different,  unless  the  simple  factors  are  all 
different.     Thus  the  expression, 

.r3  —  18a3  +  lOlz  -  180  =   (x  -  0)  (x  -  5)  (x  -  4) 

may  be  regarded  as  the  product  of  three  binomial  factors,  or 
as  the  product  of  any  one  of  these  factors  and  a  function  of 
the  second  degree  produced  by  the  remaining  two  factors. 

41.  By  this  proposition  we  find  that  every  rational  integral 
function  of  x  may  be  regarded  as  the  product  of  as  many  bi- 
nomial factors  as  it  has  dimensions.  But  to  resolve  a  given 
function  into  its  component  factors,  we  require  to  know  the 
roots  of  that  function  equated  to  zero,  ?.  e.,  the  resolution  of  a 
function  of  n  dimensions  into  factors,  requires  the  solution  of 
an  equation  of  the  nih  degree.  The  converse  operation,  how- 
ever, of  composing  a  function  from  given  simple  elements,  or 
constructing  an  equation  that  shall  have  for  its  roots  given 
quantities,  is  very  easily  accomplished.  The  roots  au  a2,  (h> 
&c,  being  given,  we  form  the  binomial  factors  (x— -«i),  (x— «_>)> 
(x— %),  &c,  the  product  of  which,  equated  to  zero,  is  the 


GENERAL    PROPERTIES   OF   EQUATIONS.  27 

equation  required.  It  is  required,  for  example,  to  form  the 
equation  the  roots  of  which  shall  be,  7,  5,  —3,  3-f-V— 2> 
3  _V^. 

The  product  of  the  binomial  factors  found  from  these  is, 

(x—1)  (x—5)  (z+d)  (x—S  +  V-^)  (a— 3— V^i) 
Multiplying  out,  and  equating  the  result  to  zero,  we  have, 
X5  _  15^4  +  64^8  +  !2^2  _  041^-  +  1155   =  0, 
which  is  the  required  equation. 

EXERCISES. 

1.  The  roots  of  x*  —  x*  —  35^2  +  129a;  -  126  =  0,  are 
3,  3,  2,  —7;  express  the  first  member  as  the  product  of 
binomial  factors. 

2.  The  roots  of  x5  —  6x*  —  37^3  +  lOGz2  +  456^  +  320 
=  0,  are  8,  5,  —4,  —2,  —  1 ;  express  the  first  member  as 
the  product  of  binomial  factors. 

3.  Form  the  equation  whose  roots  are   7,  4,  —11. 

4.  Form  the  equation  whose  roots  are   5,  4,  2,  1. 

5.  Form  the  equation  whose  roots  are   7,  3,  — 4,  — 10. 

6.  Form  the  equation  whose  roots  are  10,  8,  5,  and  1±  v-3. 

42.  Prop.  II. — An  equation  with  real  coefficients  lias  its 
imaginary  roots,  if  any,  in  conjugate  pairs. 

Let    f(x)  =  Cn  x»  +  CU  x»-1  +  . . . .  C2  x2  +  Cx  x  +  C0  =  0, 

where  Cn,  Cn.u. .  .C0,  are  all  real,  have  a  root  a  +  b  V—  1> 
then  is  a  —  b  V—  1   also  a  root. 

Let  a  +  b  V—  1  be  substituted  for  x  in  the  equation,  then 
(26)  we  obtain  a  result  of  the  form  P  +  QbV—  1,  where 
P  and  Q  contain  only  even  powers  of  b  V  —  1-  For,  when 
an  expression  (a  -j-  bV—  1)"  is  expanded  by  the  Binomial 
Theorem,  the  even  powers  of  b  V—  1  give  rise  to  real  quan- 
tities, so  that  only  odd  powers  of  b  are  affected  by  the  symbol 
V—  1 ;  and,  as  the  coefficients  of  f(x)  are  snpposed_to  be 
real,  the  symbol   V—  1   cannot  occur  in  f(a  +  bV—  1   ex- 


28  ALGEBRAICAL    EQUATIONS. 

cept  with  odd  powers  of  b.  Therefore,  if  P  +  Qby/ —  1 
denote  the  value  of  f(x)  for  a  -f  a  +  b  V—  l,jwe_can  obtain 
the  value  of  the  same  function  for  x  =  a  —  bV—  1  by  simply 
changing  the  sign  of  b  in  the  former  result.     Thus  if 

f(a  +  b  V--I)  =  P  +  QbV^h    then 

But  if  «  +  5  V—  1  be  a  root  of  the  equation  f(x)  =  0, 
we  have 

P+   e&A/^l    =    0, 

and  as  no  part  of  P  can  be  cancelled  by  Qb  V—  1,  this  result 
requires  that  P  =  0,  and  also  Q  =  0.     Hence,  also, 

P-  QbV^l  =  0, 
that  is,  a  —  b  V—  1  is  also  a  root  of  f(x)  =  0. 

43.  Cor.  1.  —  Every  function  of  an  even  degree  may  be 
regarded  as  the  product  of  real  quadratic  factors,  whatever 
be  the  character  of  the  roots.  For  if  the  function,  equated  to 
zero,  have  a  root  a  -f  b  V —  1,  b  not  being  zero,  it  must  also 
have  its  conjugate  a  —  bV—  1  as  a  root.  The  function  must 
therefore  be  divisible  by  (a  —  a  +  by/ —l) \x  —  a  —  bV—  l) 
=  a2  —  2ax  +  a2  +  b2,   a  real  quadratic  factor. 

44.  Cor.  2. —  By  a  course  of  reasoning  similar  to  that 
employed  above,  it  may  be  shown  that,  if  an  equation  have  a 
root  of  the  form  a  +Vb,  it  must  also  have  a  root  a — y/b. 

EXERCISES. 

1.  One  root  of  a4  —  27a2  +  90a  —  36  =  0  is  d  +  V^3; 
find  the  remaining  roots. 

2.  One  root  of  a4  +  5a3  —  25a2  —  140a  —  26  =  0  is 
—  5  +  y/ —  1 ;   find  the  remaining  roots. 

3.  One  root  of  6a4  +  37a3  +  53a2  -  89a  —  30  =  0  is 
i(1  —  */ —  11) ;  find  the  remaining  roots. 

4.  One  root  of  5a4—  3a3-f  6a2—  x+3  =  0  is  J(l  +  V^3); 
find  the  remaining  roots. 


GENERAL   PROPERTIES   OF   EQUATIONS.  29 

5.  One  root  of  x4  —  43.r2  —  2±x  +  108  =  0  is  G- 6457513 
=  4  4-  a/7  ;  find  the  remaining  roots. 

45.  Prop.  III.  —  To  determine  the  relations  that  exist 
between  the  coefficients  and  the  roots  of  an  equation. 

Let    f(x)  =  xn  +  C„_! xn~l  + C2a?  +  Cvx  +  Cr0  =  0. 

It  has  been  proved  that  if  «1?  a2,  a&9.  ,.an9  be  the  roots 
of  f(x)  =  0,  then 

/(z)  =  (z— ad  {x—a2)  (x—a3) ....  (x—an). 

By  actual  multiplication  we  find 

(x— «i)  (a; — «2)  =  ^2  —  («i  +  a2)a;  +  aiflfc, 
(a?— r/i)  (x— a2)  (x— a3)  =  a3  —  («!-{-  «2 + «3)  ^2 

+  (ala2+alas+  a2a^x  —  ava2(h, 
(z— ai)  (x—ch)  (x—a3)  (x—aA)  =  x*  —  (av  +  a2  +  «3  +  aja? 

+  (#i#2  +  ^i«3+  axa4-\-  a2a3+  a2a4  +  a3a4)a? 

—  {aia2a$+ aia2a±+ aia2,a±  +  a2a$a4)  x  +  ala2a$a±. 

In  these  results  we  observe  the  following  laws  to  hold : 

I.  The  number  of  terms  in  the  product  on  the  right  hand 
is  greater  by  one  than  the  number  of  binomial  factors. 

II.  The  exponent  of  x  diminishes  by  unity  in  each  term, 
from  the  first,  where  it  is  the  same  as  the  number  of  binomial 
factors,  to  the  last,  where  it  is  zero. 

III.  The  coefficient  of  the  first  term  is  unity,  that  of  the 
second  term  is  the  sum  of  the  second  letters  in  the  binomial 
factors,  that  of  the  third  term  is  the  sum  of  the  products  of 
these  letters  taken  two  at  a  time,  that  of  the  fourth  term  is 
the  sum  of  the  products  of  these  letters  taken  three  at  a  time, 
and  so  on ;  finally,  the  last  term  is  the  product  of  all  these 
letters. 

We  now  proceed  to  prove  that,  if  these  laws  hold  good  for 
(n — 1)  factors,  they  must  hold  good  for  n  factors,  and  thus 
universally.     Let  us  suppose  that 

(x— a^  (x—a2)...(x—an-i)  —  xn~l  +  SiXn~2  +  S2x"-?>  +  . . .  S„.i 


oO  ALGEBRAICAL  EQUATIONS. 

where  ft  denotes  the  sum  of  the  letters  - ax ,-a2 ,  -a,,.  .  .-aH.x , 
"      ft        "  "        "      products  of  these  letters 

two  by  two, 
"      ft        "  "        "      do.  do.        three  by  three. 

"    ft,_,      a  "        "      product  of  all  the  (n-1)  letters. 

Multiplying  both  sides  of  this  identity  by  another  factor 
x  —  an ,  the  result  will  be, 

(x—ai)  (x—a2). .  .(x—an)  =  xn  +  Si—an)xn-1 

+  (#2— /SiaB)af-2+  • ./— #«-i«"- 

Here  it  is  obvious  that  the  first  and  second  laws  still  hold. 
As  regards  the  coefficients,  that  of  the  first  term  is  still  unity. 

that  of  the  2d  term,  Si—an     =  —  alf  —a2,  —%,....  —  an 

=  the  sum  of  all  the  letters,  — «, , 

—a2, —  an. 

"     3d     «      ft  -  ft  aa  =  S2  +  an  (a, + (h + (h+  .  • .  +  ««) 

=  the  sum  of  the  products  of  all 

the  letters  two  by  two. 

"     4th   "      ft  -  ft  «„  =  ft  -  an  (a,  a2  +  a  >  r/3  +  . . . .     ) 

=  the  sum  of  the  products  of  all 

the  letters  three  by  three. 

"        "     wth    "      —  ft_i  an  =  the  product  of  all  the  letters. 

If,  then,  these  laws  hold  good  for  (n  —  1)  factors,  we  see  that 
they  hold  good  for  n  factors.  But  we  know  that  they  hold 
good  for  four  factors ;  they  must,  therefore,  hold  for  five  factors, 
and  so  on  for  any  number  of  factors. 

Therefore,  a{ ,  a2 ,  a5 , aa  being  the  roots  of 

f(x)  =  xn+  C„_i X"1  +  C„_2xn-2+  . . .  Cxx+  G0=  0,  w^e  have  also 
f(x)  =  (x—cii)  (x—a2)  (x—as) (x— a,)  =  0 

=  af  +  ft  a""1  +  ft.?"-2  +  . . .  .S^x  +  S*  =  0, 

and  equating  coefficients  of  like  powers  of  x,  we  have 

C„„i  =  ft  ,     C„_2  =  ft ,  .  .  .  .  Ci  =  ft_i ,     C0  =  ft , 

that  is,  the  coefficient  of  the  second  term  is  equal  to  the  sum 
of  the  roots  with  their  signs  changed,  the  coefficient  of  the  third 


GENERAL   PROPERTIES   OF   EQUATIONS.  31 

term  to  the  sum  of  the  products  of  every  two  of  the  roots  with 
their  signs  changed,  and  so  on,  the  coefficient  of  the  rih  term 
being  equal  to  the  sum  of  the  products  of  every  (r— 1)  of  the 
roots  with  their  signs  changed,  the  last  term  Icing  the  product 
of  them  all  taken  with  changed  signs. 

We  may  also  enunciate  these  relations  thus : 

—  C',_!  =  sum  of  the  roots  ;  Cn.2  —  sum  of  the  products  of 
every  two  of  the  roots ;  —  £„_3  =  sum  of  the  products  of 
every  three  of  the  roots;  and  generally  (  —  iyCn.r  =  sum  of 
the  products  of  every  r  of  the  roots. 

46.  It  is  obvious  that  if  the  coefficient  of  the  first  term  be 
other  than  unity,  we  must  take  the  other  coefficients  divided 
by  that  leading  coefficient  when  we  wish  to  obtain  the  sum  of 
the  roots,  &c. 

47.  As  these  relations  between  the  roots  and  coefficients 
furnish  n  independent  equations  involving  the  roots,  it  might 
naturally  be  supposed  that  by  eliminating  n  —  1  of  the  roots 
we  should  obtain  an  equation  from  which  to  determine  the 
remaining  root.  By  performing  the  elimination,  we  do,  in 
fact,  obtain  such  an  equation,  but  find  that  we  have  merely 
reproduced  the  original  equation  with  another  symbol  for  the 
unknown  quantity. 

Let  a,  b,  c,  for  example,  denote  the  roots  of  the  cubic 

xs  +  px2  +  qx  -f-  r  \  =  0, 
then  we  have 

—  p  =  a  +  b  +  c,     q  =  ab  +  ac  +  be,     —r  =  abc. 

To  eliminate  b  and  c  between  these  equations,  we  multiply 
the  first  by  a2,  the  second  by  a,  and  add  in  the  third ;  we  thus 
obtain, 

a3  +  pa2  -f-  qa  +  r  =  0. 

The  necessity  of  this  result  is  readily  perceived  when  we 
consider,  that  of  the  three  symbols  a,  b,  c,  any  one,  as  a,  repre- 
sents any  one  of  the  three  roots  without  distinction,  and  the 
equation  in  a,  which  we  obtain  by  eliminating  b  and  c,  must 
allow  of  three  values  for  a,  that  is,  must  be  a  cubic.     Yet, 


32  ALGEBRAICAL  EQUATIONS. 

though  these  relations  between  the  roots  and  coefficients  do 
not  enable  ns  to  determine  the  roots,  they  do  enable  ns  to 
discover  many  important  properties  of  roots,  as  will  be  seen 
in  subsequent  chapters. 

48.  The  following  proposition  is  independent  of  the  prin- 
ciple that  every  equation  has  a  root,  yet  it  is  most  conven- 
iently inserted  here,  as  a  lemma  to  a  useful  corollary  from  the 
preceding  proposition. 

Prop.  IV.  —  An  equation  in  wlricli  the  first  coefficient  is 
unity,  and  the  remaining  coefficients  integers,  cannot  have  a 
rational  fractional  root. 

Let        x*  +  C'(_i£"-1+ 02ar  +  C\x  +  C0  —  0 

be  such  an  equation  ;  and,  if  possible,  let  it  have  as  a  root  the 

rational  fraction  y ,  a  and  b  being  prime  to  each  other.     By 

a 
substituting  j  for  x,  and  multiplying  throughout    bn~l,  we 

have, 
an 
J 


+  Cn.ian-l+. . .  +  C2a2bn-Z+  Cxabn-2+  CQbn~l  =  0. 


an 
If  we  transpose  the  first  term  —  to  the  right  hand,  we  have 

the  remaining  terms,  which  must  form  an  integral  expression, 
since  a  and  b,  and  all  the  coefficients,  are  integers,  equal  to 

—  -j-,  which  is  not  an  integer.     This  is  impossible,  therefore  , 

cannot  be  a  root  of  the  proposed  equation. 

49.  Such  an  equation  may,  however,  have  incommensurable 
real  roots,  or  imaginary  roots  of  the  form  a  +  b  V —  1,  in 
which  a  and  b  are  incommensurable.  For  example,  the  equa- 
tion 

X*  +  5^2  _  ioz  _  8  =  0 

has  a  root  x  =  %.     Dividing  by  x  —  2,  we  obtain 

x*  +  <7x  +  4  =  0, 

which  has  for  roots  x  ==  —  J  (7  ±  Vo3),  which  are  fractional, 
but  not  rational. 


GENERAL   PROPERTIES   OF   EQUATIONS.  33 

50.  Cor. —  Since  (45)  the  final  term  is  the  product  of  all 
the  roots,  it  follows,  by  the  present  proposition,  that  in  any 
equation  having  unity  for  its  leading  coefficient,  and  the  re- 
maining coefficients  integers,  all  the  rational  roots  must  be 
integral  factors  of  the  final  term.  Also,  if  this  term,  C0,  be 
a  prime  number,  the  equation  cannot  have  rational  roots,  if 
±  1?  or  ±  C0,  be  not  roots. 

The  equation  x5  —  13a;2  -f  9x  —  11  =  0,  for  example,  must 
(12)  have,  at  least,  one  real  root ;  but  since,  upon  trial,  we 
find  that  neither  ±  1?  nor  ±  11,  is  a  root,  we  know  that  it  has 
no  rational  root. 

51.  Since,  as  will  presently  be  shown,  we  can  always  trans- 
form an  equation,  having  a  leading  coefficient  other  than 
unity,  into  one  having  unity  for  leading  coefficient,  processes, 
that  will  readily  suggest  themselves,  have  been  deduced  from 
the  above  property,  by  which  we  can,  in  any  case,  discover 
whether  an  equation  has  rational  roots.  Such  roots,  however, 
occur  rarely  in  equations  that  actually  present  themselves  for 
solution  ;  and,  when  they  do  occur,  will  necessarily  be  discov- 
ered in  the  course  of  the  process  of  analysis  hereafter  to  be 
explained.  It  is,  therefore,  not  necessary  to  make  a  special 
search  for  rational  roots.  The  following  easy  examples  given 
by  way  of  illustration,  may  be  solved  by  the  guidance  of  the 
property  given  in  Art.  50,  and  by  means  of  the  process  illus- 
trated in  Art.  21. 

EXERCISES. 

Find  the  rational  roots  of  the  following  equations : 

1.  x5  —  Gx2  +  10a?  —  3  ==  0. 

2.  Xs  —  Ylx  —  40  =  0. 

3.  tf  -  4^  -  56a?  +  217z  +  28  =  0. 

4.  x4  —  2x*  —  18z2  +  x  +  70  =  0. 

5.  x5  —  37z2  —  921a:  —  918  =  0. 

52.  Definitions. — A  Variation  is  a  change  of  sign  in 
passing  from  one  term  to  another ;  a  Permanence  is  the  con- 
tinuation of  the  same  sign  in  two  consecutive  terms. 


34  ALGEBRAICAL    EQUATIONS. 

Thus,  in  the  function, 

5^7  _  7^6  _  135  +  ^  _  17^  +  3>r2  +  qx  +  10; 

there  are  four  variations  and  three  permanences ;  the  sum  of 
the  number  of  variations,  and  of  the  number  of  permanences 
being  equal  to  the  number  expressing  the  degree  of  the  func- 
tion, as  will  evidently  be  the  case  in  a,  function  of  any  degree. 
By  changing  x  to  —  x,  the  function  becomes 

_  5*7  _  7^6  +  13a*  +  xi  +  17a*  +  3x2  -  fa  +  10, 

the  variations  being  changed  to  permanences,  and  vice  versa. 

53.  Prop.  V. — An  equation,  complete  or  incomplete,  cannot 
have  more  positive  roots  than  variations  of  sign,  and  a  com- 
plete equation  cannot  have  more  negative  roots  than  it  has 
permanences  of  sign. 

Let  the  signs  of  the  proposed  function  be,  for  example, 

+ -f  —  +  +  + +; 

we  shall  show  that  upon  multiplying  by  a  factor  x  —  a.  there 
will  be  in  the  result  at  least  one  more  change  of  sign  than  in 
the  original  function.  For,  having  to  multiply  by  a  binomial 
whose  first  sign  is  positive,  and  the  second  negative,  we  obtain, 
giving  merely  the  signs  that  occur  in  the  process, 

4-    —    —    +     —     +     +     +    —    -7    + 
—     +     +     —     +     —    —    —    +     +    — 


+'—    =F     +     —    +     ±±— ;=£-+    — 

writing  the  double  sign  in  the  result  wherever  the  sign  of 
any  term  is  ambiguous. 

Upon  comparing  the  series  of  signs  in  the  result  with  the 
series  in  the  original  function,  we  find 

(1).  For  every  group  of  permanences  there  is  a  correspond- 
ing group  of  ambiguities. 

(2).  The  signs  before  each  ambiguity,  or  group  of  ambigui- 
ties, are  contrary. 

(3).  A  final  sign  is  superadded,  which  is  necessarily  contrary 
to  the  final  sign  of  the  original  function. 


GENERAL   PROPERTIES   OF   EQUATIONS.  35 

Hence,  taking  the  most  unfavorable  case,  that  in  which  all 
the  ambiguities  are  of  the  same  sign,  we  may  by  (2)  take  the 
upper  signs  without  affecting  the  number  of  permanences ; 
then  the  signs  of  the  result,  leaving  out  the  last,  are  the  same 
as  those  of  the  original  function,  and  (3)  this  last  introduces 
an  additional  variation.  Thus  there  is  at  least  one  more  varia- 
tion than  in  the  original. 

Supposing,  therefore,  the  product  of  all  the  factors  corres- 
ponding to  the  negative  and  imaginary  roots  of  an  equation  to 
have  been  formed,  each  multiplication  by  a  factor  correspond- 
ing to  a  positive  root  will  introduce  at  least  one  change  of 
sign ;  that  is,  an  equation  cannot  have  more  positive  roots 
than  variations  of  sign. 

To  prove  the  second  part  of  the  rule,  we  have  merely  to  put 
—  x  for  x  in  the  proposed  equation  ;  then,  if  the  equation  be 
complete,  permanences  are  changed  to  variations,  and  vice  versa. 
The  transformed  equation  cannot  have  more  positive  roots 
than  variations,  that  is,  the  proposed  equation  cannot  have 
more  negative  roots  than  permanences. 

54.  Cor. —  Hence,  when  the  roots  of  an  equation  are  all 
real,  the  number  of  variations  is  exactly  equal  to  the  number 
of  positive  roots,  and  the  number  of  permanences  to  the  num- 
ber of  negative  roots.  For  if  m  and  r  be  respectively  the 
number  of  positive  and  negative  roots,  and  m'  and  r'  respect- 
ively the  number  of  variations  and  permanences;  then,  since 
m  +  r  =  m'  +  r\  each  being  equal  to  the  degree  of  the  equa- 
tion, and  m  cannot  exceed  m',  nor  r,  r',  Ave  must  have 
m  =  m',  and  r  =  r'. 

55.  This  important  theorem,  generally  called  Descartes's 
Rule  of  Signs,  from  the  name  of  its  discoverer,  is  included  as 
a  particular  case  in  Fourier's  Theorem,  (180),  in  connection 
with  which  some  useful  deductions  will  be  given,  which  are 
usually  deduced  as  corollaries  from  this  proposition. 


36  ALGEBRAICAL  EQUATIONS. 


CHAPTEK    IV. 

TRANSFORMATION    OF    EQUATIONS. 

56.  Without  knowing  the  roots  of  an  equation,  it  is  in 
our  power  to  derive  from  it  various  other  equations  the  roots 
of  which  have  given  relations  to  those  of  the  proposed  equa- 
tion. These  transformations  are,  some,  useful  in  preparing 
the  equation  for  solution  by  reducing  it  to  a  more  convenient 
form,  and  some  are  necessary  to  the  actual  solution.  The 
propositions  of  the  present  chapter  comprise,  by  no  means  all, 
but  the  most  useful  and  simple  cases  of  the  general  problem, 
to  transform  an  equation  into  another  the  roots  of  which  shall 
be  any  given  functions  of  those  of  the  proposed  equation. 

57.  Prop.  I.  —  To  transform  an  equation  having  negative 
or  fractional  exponents  into  another  having  only  positive  inte- 
gral exponents. 

Let  the  proposed  equation, 

f(x)  =  ax-1'1  +  lxr  +  cx~s+ lex'1  +  5  =  0, 

have  negative  exponents,  integral  or  fractional.  Suppose  —  s 
to  be  the  numerically  greatest  negative  exponent ;  then,  mul- 
tiplying throughout  by  x%  we  have 

f(x)xe  =  ax°~m+  bxr+'+  c  + fcx  *-*+&;•  =  0, 

in  which  no  negative  exponents  occur.     It  is  obvious  that  the 
roots  of  f{x)  =  0   are  not  changed  by  this  transformation, 
since,  if  f(x)  =0  for  a  certain  value  of  x,  then  also  f(x)x* 
=  0  for  the  same  value,  and  vice  versa. 
Again,  to  transform  the  equation, 

?/i  p  r  t 

f(x)  —  ax"  +  lx*  +  cx~s+ kx*+l  =  0, 

into  another  having  only  integral  exponents,  assume  y'x  =  x, 


TRANSFORMATION     OF    EQUATIONS.  37 

where  \i  =  n  qs . . . v,  or  the  L.  0.  M.  of  the  denominators 
of  the  exponents.  Substituting  yv-  for  x  in  the  proposed, 
we  have 

f(y)  =  aym'  +  by?' +  cff '+  ....  +/tyr+  J  =  0, 
where  m  =  —  x  ju,  p'  =  -   x  ^,  &c,  are  integers. 

The  relation  between  the  roots  of  f(y)  =  0  and  f(x  =  0 
is  2/  =  V#,  that  is,  any  root  of  the  transformed  equation 
in  y  must  be  raised  to  the  fith  power  to  be  a  root  of  the  pro- 
posed equation. 

58.   These  results  may  be  formulated  in  the  following 

Rule. — To  free  an  equation  from  negative  exponents,  mul- 
tiply out  by  the  reciprocal  of  whatever  power  of  x  occurs  with 
the  numerically  greatest  negative  exponent.  To  free  an  equa- 
tion from  fractional  exponents,  substitute  in  the  proposed 
equation  y-  for  x,  ivhere  \i  is  the  L.  C.  M.  of  the  denominators 
of  the  exponents. 

Ex.  I.  —  It  is  required  to  transform  the  equation, 
3aT*  -f  17z*  —  32ar-a  +  9x  —  8  =  0, 

into  another  having  only  positive  integral  exponents. 
Multiplying  throughout  by  a4,  we  have 

Sxi  +  17^  —  32  +  9x5  —  Sx*  —  0  ; 

from  this,  by  substituting  if  for  x,  we  obtain 

3if  +  17?/28  —  32   +  90?/30—  Sty™  =  0, 
or,  9y*>  +  17y®  —  8y*-\-    3y9  —  32    =0. 

EXERCISES. 

Transform  the  following  equations  into  others  having  only 
positive  integral  exponents: 

1.    x*  +  3ar*  +  5x~*  -7  =  0. 

2.  I  +  A_?_*2==o. 

X2  X5  X 


38  ALGEBRAICAL  EQUATIONS. 

3.  3x  —  hx%  -f  7a;*  —  lx-z  =  0. 

4.  2xi  +  y&  +  3^~3  =  0. 

5.  *^I  =  1  +  **. 


G.    yi  +  a3  =  3a?—  or3. 

7.  vr^  =  v7*  —  i. 

59.  Prop.  II.  —  Jb  transform  an  equation  into  another, 
whose  roots  shall  be  those  of  the  proposed  equation,  each  mul- 
tiplied by  a  given  quantity. 

Let   f{x)  =  Cnxn  +  Cn_xxn-l+ C2x-  +  C\x  +  Q,  =  0 

be  an  equation  which  we  wish  to  transform  into  another,  the 
roots  of  which  shall  be  m  times  as  great  as  those  of  f(x)  =  0. 

11  .  u 

Suppose  y  =  mx,  then  —  =  x.      Replacing  x  by  —  in  the 

proposed  equation,  we  have 

aJS)\  a.-,  [^'\  . . . .  +  cM+  oi!)  +  c0  =  o. 

\ml  \mj  \ml  \mj 

Multiplying  throughout  by  mn,  we  have 

/  (y)  =  C„  if  +  m  Cn.x  yn~l  +  . . .  +  mn~2  C2  y2  +  ??z"-]  C\  y  +  mn  C0 = 0, 

an  equation  whose  roots  are  each  m  times  as  great  as  those 
of  f(x)  =  0.  Hence  we  have  the  following  rule,  before 
applying  which  we  complete  the  equation  by  supplying  the 
place  of  absent  terms  by  zero's. 

Rule. — To  transform  an  equation  f(x)  =  0  into  another, 
fiyij)  =  0,  the  roots  of  ivhich  shall  be  m  times  as  great,  replace 
x  by  y,  and  multiply  the  coefficients,  beginning  with  the  second, 
by  m,  m2,  m% ....  mn. 

Ex.  1.  — Transform     re4  —  19a?  +  49a2  +  293  —  17   —   0 
into  an  equation  having  its  roots  three  times  as  great. 
Here  m  =  3 ;   proceeding  by  the  rule,  we  obtain 

f  —  bit/  +  441?/2  +  7SSy  —  1377  =  0. 


TRANSFORMATION    OF   EQUATIONS.  39 

Ex.  2.  — Transform    x4  —  5L?2  +  l?6z  —  789   =   0     into 
an  equation  having  its  roots  one-tenth  as  great. 
First,  completing  the  equation,  we  have, 

x*  +  Oz3  —  51a;2  +  176a  —  789  =  0  ; 

then  proceeding  by  the  rule,  and  multiplying  by  powers  of  TV, 

we  obtain 

yi  _  .51^2  +  >176y  _  .0789  =  0. 

60.  The  chief  use  of  this  transformation  is  to  enable  us  to 
have  unity  as  leading  coefficient  without  introducing  fractional 
coefficients ;  or,  to  clear  the  equation  of  fractional  coefficients, 
without  affecting  the  leading  term  with  any  coefficient  other 
than  unity. 

Ex.  1.  —  Thus,  if  we  have  the  equation 

5^4  _  w  +  11^2  +  5x  _  3  _  o, 

by  taking  m  —  5,  we  have 

5^4  _  7  x  5^3  +  ii  x  52^2  +5x53y-3x54=  0, 
or,         if  —  If        -f  55#2        +  125y     —  375     =  0. 

Ex.  2.  —  If  we  have  to  clear  of  fractional  coefficients 

*  +  3X*  ~  25*2  +  60^  +  120  =  °' 

we  have  to  consider  what  m  must  be,  so  that  its  powers  will 
contain  the  denominators  of  the  fractional  coefficients. 
By  taking  m  =  30,  and  proceeding  by  the  rule,  we  obtain 

f  +  \  (30)  f  -  §  (30)y  +  1  (30ft  +  ^  (30)*  =  0, 
or,    #4  +  40#s        —  396 f        +  3150?/       +  87750      =  0. 

EXERCISES. 

Transform  the  following  equations  into  others  having  the 
leading  coefficient  unity,  and  the  remaining  coefficients 
integers  : 

1.  5a;3  —  Sx2  —  \\x  —  20  =  0. 

2.  W  —  54.T2  +  30  =  0. 


40  ALGEBRAICAL    EQUATIONS. 

3.  s8  -  23.54a;2  +  36.7a;  -  .745  =  0. 

4.  3^  +  ^-^+1  =  0. 

5.  ^_l-z2-  |a  +  3   =  0. 

61.  Prop.  III. — To  transform  an  equation  into  another, 
ivhose  roots  are  those  of  the  proposed  equation,  with  the  con- 
trary sign. 

Let   f(x)  =  Cnxn  +  Cn.xxn-l+  ....C2x2-j-C1x-\-C0  =  0. 

The  transformation  of  this  equation  into  another,  having 
roots  numerically  the  same,  but  with  contrary  sign,  is  merely 
a  particular  case  of  the  preceding  transformation.  We  replace 
x  by  y,  and  take  m  =  —  1,  that  is,  multiply  the  coefficients, 
beginning  at  the  second,  by  the  successive  powers  of  —  1, 
which  are  alternately  negative  and  positive ;  thus  we  obtain 

Onfr  -  CUr*+  •  •  •  •  ± 02f  =F  CLy  ±  C0  =  0 
as  the  required  equation. 

Bule.  —  The  signs  of  the  alternate  terms  of  a  complete  equa- 
tion being  changed,  the  signs  of  all  the  roots  will  he  changed. 

Ex.  1.  — Let 

X5  +  na4  _  23z3  —  47a:2  +  S2x  -  57  =  0 ; 

then,  replacing  x  by  y,  and  changing  the  signs  of  alternate 
terms,  we  have 

y5  _  nyi  _  23?/3  +  47#2  +  32y  +  57  =  0, 

an  equation  whose  roots  differ  from  those  of  the  proposed  only 
in  having  the  contrary  sign  ;  i.  e.,  y  =  —x. 

63.  When  an  equation  is  incomplete,  we  may  omit  to 
complete  it  by  supplying  zero  coefficients,  remembering,  when 
replacing  x  by  y,  to  change  the  signs  of  those  terms  contain- 
ing odd  or  even  powers  of  x,  according  as  the  exponent  in  the 
first  term  is  even  or  odd.     Thus, 

Ex.  2.  — Let  f(x)      =  x*  +  34a3  -  96a;2  +  5  =  0,    then 
f{—x)  =  a;6  -  34a-3  -  96a?  +  5  =  0. 


TRANSFORMATION    OF    EQUATIONS.  41 

The  equation  whose  roots  differ  from  those  of  f(x)  =  0  in 
sign  only,  is  often  conveniently  denoted,  as  above,  by  /(—  x) 
=  0. 

63.  This  simple  transformation  is  very  useful,  as  we  are 
thus  enabled  to  confine  our  attention  to  the  discovery  of  rules 
for  the  determination  of  positive  roots,  it  being  always  in  our 
power  to  change  negative  into  positive  roots. 

EXERCISES. 

Transform  the  following  equations  into  others,  whose  roots 
are  numerically  the  same,  with  contrary  sign. 

1.  a?  —  17a:2  —  53a:  +  73  =  0. 

2.  x*  —  23a:3  +  130a:2  —  305a:  +  96  =  0. 

3.  x5  +  37a^  —  Ilia:2  —  546  =  0. 

4.  6.T5  —  41a?  —  65a:2  +  239a:  +  426  =  0. 

5.  8a?  +  57a:3  —  172a:2  —  306a:  +  150  =  0. 

6.  a*  -  73a?  —  54a:4  +  29a:2  —  13  =  0. 

Note  this  last  example,  and  explain  why  in  this  case  f(x) 
has  the  same  signs  as  /(—  x). 

64.  Prop.  IV.  —  To  transform  an  equation  into  another, 
whose  roots  are  the  reciprocals  of  those  of  the  proposed  equa- 
tion. 

Let  the  proposed  equation  be 
f{x)  =  Onxn  +  CUar*+  ....C2x*+  dx  +  Co  =  0. 

Assume  y  =  -,  and  substitute  -  for  x  in  f(x) ;  then 
x  y 

Multiply  throughout  by  yn,  and  reverse  the  order  of  terms ; 
then 

Cor  +  Citr'1  +  c2r~2+ ....  CUy  +  Cn  =  0, 

an  equation  in  which,  since  y  =  -,  the  roots  are  the  re- 

x 

ciprocals  of  those  of  the  proposed  equation. 


42  ALGEBRAICAL  EQUATIONS. 

Kule.  —  The  coefficients  of  the  equation  of  the  reciprocals 
of  the  roots  are  those  of  the  proposed,  written  in  reverse  order. 

Ex.  1. — Find  the  equation  involving  the  reciprocals  of  the 

roots  of 

5^4  _  7a*  +  lla*  +  23x  -3  =  0. 

Proceeding  by  the  rule,  and  replacing  x  by  y,  we  have 

-  3y*  +  2Sy*  +  llif  -  Ty  +  5  =  0, 
or,  3if  —  23y*  —  lly*  +  7#  —  5  =  0, 

which  has  for  roots  the  reciprocals  of  the  roots  of  the  pro- 
posed. 

65,  This  transformation  is  chiefly  of  use  in  Lagrange's 
Method  of  Solution,  and  in  Budan's  Method  of  Analyzing- 
Equations,  neither  much  used. 

EXERCISES. 

Find  the  equations  having  as  roots  the  reciprocals  of  those 
of  the  following  equations : 

1.  3z3  —  5a;2  +  20z  —  11  =  0. 

2.  W  —  54z2  —  12x  —  15  =  0. 

3.  x5  —  1W  +  32^2  +  7x  —  23  =  0. 

66.  Prop.  V. —  To  transform  an  equation  into  another, 
the  roots  of  tuhich  shall  oe  the  squares  of  the  roots  of  the  pro- 
posed equation. 

Let    f{x)  =  Cnxn  +  Cn^xn~l+  ....  C2x2+Cxx+  C0  =  0  ; 

the  terms  may  be  arranged  in  the  following  order, 

f(x)  =  (Gnxn  +  Cn_2x-*  +  . . . .  C2a?  +  C0) 

the  terms  containing  even  powers  of  x  beiug  collected,  as 
here,  with  Cnxn  when  ^  is  even,  otherwise  the  odd  powers. 

Similarly,  since  in  f(—x)  the  signs  of  the  alternate  terms 
are  contrary  to  those  of  the  corresponding  terms  in  f(x), 
we  have 


TRANSFORMATION    OF    EQUATIONS.  43 

f(—x)  =  {Cnx»  +  CH.»a?-*+  . . .  C2x2  +  G0) 

-  (0,-i«B-1+^a^+  .  •  •  C^+  dx)  =  0. 
...  f(x)'f(-x)  =  (CHx»+  Cn_2xn-2+  . . .  C2x«+  C,f 

-  (C7n-i  af*+  CU^H-  •  •  •  C*x*+  Glxf=  0. 

Squaring  the  expressions  within  the  brackets,  collecting,  and 
arranging  according  to  powers  of  x,  we  have  for  f{x)f{—x) 

-(CllJ-2CH-2Cn.i  +  &c.)x2»-«+  . . .  ±(Cl2-2C2C0)x2 

=f  W  =  0, 
the  coefficient  of  (xr)2  being 

±(^_2(7,.1(7,+1  +  2a.2a-+2-2CUa-+3+  •  •  •  ±»0W  0^), 

the  upper  sign  being  taken  when  6>  stands  in  an  odd  term, 
and  vice  versa,  and  Or+m  denoting  the  nearer  of  the  extreme 
coefficients  Gn  or  CQ . 

The  equation  thus  obtained,  by  the  product  of  f(x)  and 
f(—x),  and  which  we  shall  denote  by  F(x2)  =  0,  has  for 
roots  the  squares  of  the  roots  of  f(x)  =  0,  since  for  every 
factor  (x — a)  in  f(x),  there  is  a  corresponding  factor  (x-\-a) 
in  f(—x),  and  therefore  a  factor  (x2— a2)  in  i^(^2) ;  there 
are  therefore  n  such  factors,  and  the  equation  F(x2)  =  0  has 
for  roots,  a2,  a2,  a2,. . .  .an2,  if  ai}  «2,  a$,. . .  .  «n  are  the  roots 
of/(aO=0. 

We  obtain  the  coefficients  of  F(x2)  from  those  of  f(x) 
as  follows : 

Kule. —  Given  a  coefficient  Gr  of  f(x)  =  0,  the  correspond- 
ing coefficient  of  F(x2)  =  0  may  be  found  by  talcing  from  the 
square  of  Gr  the  double  product  of  the  immediately  contiguous 
coefficients,  adding  to  the  result  the  double  product  of  the  co- 
efficients next  removed  on  each  side,  and  so  on,  as  far  as  the 
coefficients  extend  ;  to  the  final  result  we  prefix  a  positive  sign, 
if  Cr  stands  in  an  odd  term,  the  negative  sign,  if  it  stands  in 
an  even  term. 

Ex.  — Given,  a*  —  ox3  +  llz2  +  7x  —  6  =  0 ;  find  the 
equation  of  the  squares  of  the  roots. 


44  ALGEBRAICAL  EQUATIONS. 

Here  the  expressions,  which,  with  alternately  positive  and 
negative  signs  prefixed,  will  be  the  coefficients  of  F{x2),  are, 

1,     (25  -  22),     (121  +  70  —  12),     (49  +  132),   and   36 ; 
the  required  equation  is,  therefore, 

a*  _  3^6  +  179^4  _  181aa  _j_  36  _  o. 

67.  This  transformation,  as  will  be  seen  in  Chap.  X,  is 
capable  of  being  applied  to  good  purpose  in  the  analysis  of 
equations. 

EXERCISES. 

Find  the  equations  whose  roots  are  the  squares  of  the  roots 
of  the  following  equations : 

1.  a*  -  13z3  +  12z2  —  6x  —  5  =  0. 

2.  5z4  —  IW  —  23x  +  19  =  0. 

3.  x5  —  27z3  +  90a;2  -  3Gx  -  25  =  0. 

68.  Prop.  VI. —  To  transform  an  equation  into  another, 
the  roots  of  which  are  those  of  the  projjosed  equation,  each 
increased  or  diminished  by  a  given  quantity. 

Let  f(x)  =  Cnxn  +  ft-ia^M-  •  •  •  •  C*&  +  C±x  +  fl  =  0, 

be  the  given  equation.  Assume  y  =  x  —  a,  then  whatever 
value  is  assigned  to  x,  that  of  y  is  less  or  greater  according  as 
a  is  positive  or  negative.  The  required  transformation  is, 
therefore,  f(a  +  y)  —  0.     By  Art.  7  we  have 

/(«+*)  =/(«)+ fMy+\fMf+  •  •  -jizr/iWjr4 

+  ^fM)vn>  =  o, 

or,  writing  according  to  descending  powers  of  y,  and  putting 
C„  for  ±-f„(a), 

f(a  +  y)  =  g.y+j^ZI/.-.(%-'+  •  •  •  +l/2(a)f+A(a)>J 

+/(«)  =  0, 

an  equation  the  roots  of  which  are  less  than  those  of  f(x)  =  0 
by  the  quantity  a,  if  a  is  positive,  and  vice  versa. 


TRANSFORMATION    OF    EQUATIONS.  45 

69.  In  Art.  21  was  explained  and  illustrated  a  convenient 
method  for  obtaining  the  value  of  f{a).  As  intimated  iu  (22), 
we  can,  by  continuing  the  process  with  the  successive  quo- 
tients, obtain  in  succession  the  values  of  f\{a),  \f2(a)>  ^fi(a), 
&c,  which  are  the  coefficients  of  the  transformed  equation. 

Ex.  1. — Transform  the  equation 

5Z4  —  IW  +  29z2  +  12z  —  19G  =  0 
into  another  whose  roots  are  each  less  by  3. 

Writing  down  coefficients  only,  we  proceed  as  follows  : 

5  _  17   +  29    +   12  — 196     |J^_ 
15   _     6    +   69   +  243 


1st  Quot.    5—    2    +  23    +   81; +  f47  =  1st  Bern.,  or, /(3). 

15    +   39    +  186__J 

2d  Quot.     5  +  13    +   62; +267  =  2d  Kern.,  or,  fx (3). 

15    +  84 
3d  Quot.     5  +  28;  +146  =  3d  Kern.,  or,  J/2(3). 

15 

4th  Quot.    5;     43  =  4th  Kern.,  or,  ]J-/s(3). 

Writing  out  the  transformed  equation  with  these  remain- 
ders as  coefficients,  we  have 

62/  +  43#3  +  146^2  +  267?/  +  47  =  0. 

Since  we  have  taken  for  a  a  positive  quantity  3,  the  roots 
of  this  transformed  equation  are  each  less  than  those  of  the 
proposed  equation  by  3.  For  the  sake  of  exhibiting  the 
entire  quotient,  we  have  taken  down  the  leading  coefficient 
each  time  in  the  above  example ;  in  practice  the  process  is 
performed  as  in  the  following  example  : 

Ex.  2.  —  Transform  the  equation 

Sx5  +  19£3  —  27a:2  —  121a;  +  20  =  0 

into  another  whose  roots  shall  be  each  less  by  2. 

Supplying  the  absent  terms  by  zero,  we  proceed  as  fol- 
lows : 


46  ALGEBRAICAL    EQUATIONS. 


3  +  0 

+  19 

-  57 

-  21 

+  46 

6 
6 

12 
31 

62 
5 

10 
-11 

-22 
24 

6 
12 

24 
55 

110 
115 

230 
219 

6 
18 

36 
91 

182 
297 

6 
24 

48 
139 

_6 
30 
The  transformed  equation  is, 

3^5  _j_  30^4  +  139^3  +  297#2  +  219?/  +  24  =  0. 

Ex.  3.  —  Transform  the  equation 

8^  +  73z3  +  U7x2  —  125a;  +  227  =  0 
into  another  whose  roots  are  each  greater  by  5. 
Here  we  take  a  =  —  5,  and  proceed  as  before. 

8    +      ?3     +     147    —     125     +     227     1  —  5 


—40 

-165 

90 

+  175 

33 

—  18 

—35 

402 

—40 

35 

—  85 

-  7 

17 

—120 

-40 

235 

-47 

252 

—40 

-87 
The  transformed  equation  is, 

8^  _  87#3  +  252?/2  -  120y  +  402  =  0. 

70.  This  transformation  may  be  regarded  as  the  most 
important  of  those  given  in  the  present  chapter.  It  will  be 
found  that  both  the  analysis  and  solution  of  equations  may 
be  completely  effected  by  means  of  a  series  of  these  trans- 
formations. 


TRANSFORMATION    OF    EQUATION'S.  47 

EXERCISES. 

Transform  the  following  equations  into  others  whose  roots 
are  less,  by  the  numbers  placed  to  the  right. 

1.  at  —  15s8  +  73z2  -h  712*  —  85  =  0.  a  =  5. 

2.  7^  —  46*2  +  93*  —  105  =  0.  a  =  3. 

3.  II*4  -  29*3  +  56*  +7   =  0.  a  =  4. 

4.  33s  -  1W  +  57*3  -  78a:2  +  93  =  0.  a  =  -2. 

5.  S*5  +  78*2  —107a;  —  540  =  0.  a  =  —  5. 

71.  Prop.  VII.  —  7*o  transform  an  equation  into  another, 
in  which  any  assigned  term  shall  be  absent. 

This  is  merely  a  particular  case  of  the  preceding  trans- 
formation. 

If  the  roots  of  /(*)  =  0  be  diminished  by  k,  we  have 

+..../(*)=«■ 

In  order  that  the  (r+l)ft  term  in  this  may  be  absent,  we 
must  take  k  so  that  the  coefficient  of  yn'r  may  be  zero, 
i,e.,fn-r(Jc)  =0.    From  (7)  we  find  that  fn-r{k)  =  (OJf 

+  l0^-1  +  $^)°^-*+  &°-^r>  S°  that  k  Wi" 
have  to  be  determined  by  an  equation  of  the  rth  degree. 

To  take  away  the  third  term,  r  becomes  2,  and  we  find  k 
from  the  quadratic, 

GnW  +  -  CU  k  +     ,%  1V  CU  =  0. 
w  w(w— 1) 

To  take  away  the  second  term,  a  transformation  that  is 
often  found  advantageous,  we  determine  k  from  the  simple  equa- 

tion  (Life  +  -CU  =  0,  whence  we  obtain  k  =  —  -~ . 

Ex.  1.  — Transform  **  —  15*2  .+ 11*  —  13  =  0,  into  an 
equation  without  the  second  term. 

n  15 

Here  — ■£-  =  — ,  =  5,  and  we  proceed  thus  : 
nC„  3 


48  ALGEBKAICAL  EQUATIONS. 

1    - 


15 
5 

+       11 
-50 

-        13 
—  195 

I5 

-10 
5 

-39 
-25 

-208 

-5 
5 

-64 

0 

Therefore  the  required  equation  is, 

f  —  Uij  —  208  =  0. 

Ex.  2.  — Transform  5x3  —  lla?  +  S2x  —  15  =  0,  into  an 
equation  without  the  second  term. 

In  order  to  avoid  fractional  coefficients,  we  first  by  (59) 
transform  the  equation  into  another,  the  roots  of  which  are 
15  times  as  great ;  we  thus  obtain, 

ys  _  33^2  +  1440y  _  10125  _  0j 

and  proceed  as  before,  obtaining, 

*3  +  10772  +  3053  =  0. 

72.  To  take  away  any  coefficient  except  the  second  is  not 
of  any  practical  advantage ;  it  is  besides  evident  that,  in  the 
great  majority  of  cases,  h  would  be  incommensurable,  when 
determined  by  equations  of  the  second  and  higher  degrees. 

EXERCISES. 

Deprive  the  following  equations  of  their  second  terms  : 

1.  Xs  —  12x*  +  15x  —  3  =  0. 

2.  x5  +  27z2  —  32x  —  54  =  0. 

3.  4^  —  2rc2  +  21a;  +  30  =  0. 

4.  a*  -  24z3  +  42z2  -  37z  +  49  =  0. 

5.  3x*  +  22z3  —  56x  +  13  =  0. 


LIMITS    OF   THE    ROOTS    OF    EQUATIONS.  49 


CHAPTER    V. 

LIMITS    OF    THE    ROOTS    OF    EQUATIONS. 

73.  In  the  preceding  chapter  have  been  investigated 
methods  by  which  an  equation  may  be  reduced  to  a  form 
convenient  for  solution.  We  have  next  to  ascertain,  as  closely 
as  possible,  between  what  limits  in  the  numerical  scale  the  real 
roots  lie,  so  as  to  avoid  unnecessary  labor  in  the  search  for 
those  roots. 

Definition.  —  A  superior  limit  to  the  positive  roots  of  an 
equation  is  any  number  that  is  nearer  to  +  co  than  the 
greatest  of  these  roots;  an  inferior  limit,  a  number  that  is 
nearer  to  0  than  the  least  positive  root. 

A  superior  limit  to  the  negative  roots  is  any  number  nearer 
to  0  than  the  numerically  least  of  these  roots;  an  inferior 
limit,  any  number  that  is  nearer  to  —  c©  than  the  numerically 
greatest  of  these  roots. 

A  superior  limit  to  the  positive  roots  of  f(x)  =  0,  might 
also  be  denned  as  being  such  a  number  that,  when  it,  or  any 
greater  number,  is  substituted  for  x  in  f(x),  the  result  is  pos- 
itive :  for  if,  for  any  substitution,  we  obtain  a  negative  result, 
there  must  be  a  root  greater  than  the  number  that  produced 
that  result. 

In  the  following  propositions  respecting  the  limits  of  the 
roots  of  an  equation,  it  is  to  be  understood  that  the  coefficient 
of  the  first  term  is  unity,  unless  the  contrary  is  stated. 

74.  Prop.  I. —  The  greatest  negative  coefficient  of  an  equa- 
tion, taken  positively  and  increased  by  unity,  is  a  superior 
limit  to  the  positive  roots. 

Let  —  h  be  the  numerically  greatest  negative  coefficient 
in  f(x)  =  0,  which  we  shall  suppose  to  be  of  the  nth  degree, 


50  ALGEBRAICAL    EQUATIONS. 

then  any  value  of  x  that  makes  xn  —  k  (xn~l  +  xn~2 -f- x-\-l) 

X" l 

positive,  that  is,  xn  >  h ,  will,  a  fortiori,  make  f(x) 

positive.    For,  even  supposing  that  all  the  coefficients  after 

the  first  are  negative,  none  of  them  is,  by  supposition,  numer- 
al  J 

ically  greater  than  —  k.     Now  the  inequality  xn  >  k 

is  satisfied  if  x"  =  or  >  xn . -,  that  is,  if  x— 1  =  or  >  k, 

X  —  J. 

or,  x  =  or  >  k  +  1.  Hence  f(x)  is  positive  when  x  =  h  +1 
or  any  greater  number,  that  is,  k  -\-  1  is  a  superior  limit  to  the 
positive  roots. 

Ex.  1.    f(x)  =  x4  —  15x*  +  132  -  5   =  0. 

By  the  present  proposition  15  is  a  superior  limit  to  the 
positive  roots. 

75.  Prop.  II.  —  In  an  equation  of  the  nth  degree,  if  —  k 
be  the  numerically  greatest  coefficient,  and  xn~r  the  highest 
power  of  x  that  has  a  negative  coefficient,  then  *\fk  -f  1  is  a 
superior  limit  to  the  positive  roots. 

Let  f(x)  ==  0  be  the  proposed  equation.  Since  all  the 
terms  that  precede  xn~r  are,  by  supposition,  positive,  any  value 
of  x  that  makes  xn  >  k  (%*~r  +  x"-"1  + x  +  1),  that  is, 

/v.n-r+1 1 

xn  >  k. — - — —  will  obviously  make  f(x)   positive.      The 

X                                                              xn~r+1 
preceding  inequality  is  satisfied  if    xn  >  k  . ,    or    if 

X  —  JL  i 

xr~x  (x— 1)  >  k,  or  if  (x—l)r  =  or  >  k,  or  if  x  =  or  >  Jc~r 
+  1.  Hence  for  x  =  <\/k  +1,  or  any  greater  number,  f(x) 
is  positive,  that  is,  \fk  +  1  is  a  superior  limit  to  the  positive 
roots. 

Ex. — Taking  the  same  equation  we  had  above, 
o4  _  16x2  +  i$x  _  5  _.  o, 

we  have  k  =  15,  and  r  =  4  —  2  =  2 ;  therefore  \/l5  +1, 
or  5,  is  a  superior  limit  to  the  positive  roots,  a  limit  much 
closer  than  that  obtained  by  the  preceding  method. 


LIMITS    OF   THE    HOOTS    OF   EQUATIONS.  51 

76.  Prop.  III.  —  If  each  negative  coefficient,  taken  posi- 
tively, be  divided  by  the  sum  of  all  the  positive  coefficients  that 
precede  it,  the  greatest  quotient  thus  obtained  will,  increased 
by  unity,  be  a  superior  limit  to  the  positive  roots. 

Let  the  proposed  equation  be, 

f(x)  =  Cnxn  +  C^x"-1  +  C.-.x"-2  -  CUz"-3+  .- .  .Cra? 

+ . . . .  C0  =  0. 

To  demonstrate  the  proposition,  we  arrange  f(x)  in  a  form 

deduced  from  the  expression  for  the  sum  of  a  geometrical 

xm l 

series,  — — —  =  xm~l  -f-  xm  2  -f  . . . .  x  +  1,  whence  we  have, 

X  —  J. 

xm  =  (x—1)  (xm-1  +  xm~2+ x  +  1)  4- 1. 

We  therefore  transform  all  the  positive  terms  of  the  equation 
by  this  formula,  leaving  the  negative  terms  unchanged,  thus  : 

Cn   x-  =Cn(x-l)x*-l+Cn  (x-l)xn'2+Cn   (x-l)x»-*  +  ...Cn 
+  Cn.1afl-1=  Cn.x(x-l)x»-2+  Cn_l{x-l)xn-*  +  ...Cn-x 

+  Cn.2xn~2=  C>t_2(x-l)x»s  +  ...Cn-2 

-Cn_3xn-3=  -CU         ^"3 

and  so  on,  with  the  remaining  terms. 

It  is  evident  that  in  a  given  column  only  one  negative 
coefficient  can  occur,  since  for  different  negative  coefficients 
the  powers  of  x  are  different.  If  in  any  vertical  column  there 
occur  no  negative  coefficient,  that  column,  of  course,  is  posi- 
tive if  x  is  not  less  than  unity.  In  a  column,  in  which,  as  in 
that  containing  xn~z  above,  a  negative  coefficient  occurs,  we 
must,  in  order  to  obtain  a  positive  result,  have 

and  ( Cn  +  C„_i  +  •  •  •  •  C-i)  (x-l  >   Or , 

if  the  column  containing  xr  has  a  negative  coefficient  Cr ; 

that  is,  we  must  have  x  greater  than    n         " — — ^ V  1, 

C 
and  greater  than-^ Ti — {- -~ V  1. 


52  ALGEBRAICAL  EQUATIONS. 

If  x,  then,  be  taken  greater  than  the  greatest  of  these 
expressions,  that  value  of  x  will  make  all  the  vertical  columns 
positive,  and  consequently  make  f(x)  positive.  Hence  the 
greatest  of  these  expressions  is  a  superior  limit  to  the  positive 
roots  of  f(x)  =  0. 

This  is  generally  the  most  effective  of  the  three  limits  yet 
given,  and  does  not  require  the  leading  coefficient  to  be  unity. 

Ex.  1.     x5  +  13a4  —  21z3  —  69a?  +  81a  —  58  =  0. 

By  (74)  we  have   +70  as  a  superior  limit. 

By  (75)  we  have  a/69  -f  1,  or  10,  as  a  superior  limit. 

By  the  present  proposition  we  take  the  greatest  of  the  ex- 

21       ,   .,         69  58 

presses,  rT33  +  1,    j^  +  1,   and    ,  +  13  +  gl  +  1 1 

the  second  is  plainly  the  greatest,  therefore  6  is  a  superior 
limit. 

Ex.  2.    3x5  —  17a:4  +  56s8  —  78z2  +  9Sx  —  105  =  0. 

By  (74)  we  have  36  as  a  superior  limit. 
By  (75)  we  obtain  the  same  limit. 

17 

By  this  proposition  we  have    —  +  1,    or   7,   as  superior 

o 

limit. 

Ex.  3.    x5  +  W  —  156a2  —  453z  —  954  =  0.  \ 

By  (74)  we  have  955  as  superior  limit. 

By  (75)  we  have  a/954  -f  1,  or  11,  as  superior  limit. 

77.  It  is  not  generally  of  much  importance  to  obtain  an 
inferior  limit  to  the  positive  roots.  Such  a  limit  may  be  found 
by  transforming  the  equation  into  another,  whose  roots  are  the 
reciprocals  of  those  of  the  proposed  equation.  The  reciprocal 
of  a  superior  limit  to  the  roots  of  this  transformed  equation, 
will  be  an  inferior  limit  to  the  positive  roots  of  the  proposed 
equation. 

Ex.  4.    z3  +  15z2  _  37z  _  45  =  o. 
The  equation  of  the  reciprocals  is, 

45#3  +  37?/2  _  i5y  _  l  =  o. 


LIMITS    OF    THE    ROOTS    OF    EQUATIONS.  53 

A   superior  limit  to  the   positive   roots   of   this    is    (76), 
15  97  82 

4.K  ,  aw  +  1>  or  oo  '   tneref°re  Q7  is  an  inferior  limit  to  the 

positive  roots  of  the  proposed  equation. 

78.  To  find  limits  to  the  negative  roots,  we  change  the 
alternate  signs  of  the  proposed  equation  (taking  account  of 
absent  terms),  thus  obtaining  the  coefficients  of  the  equation 
whose  positive  roots  are  numerically  the  same  as  the  negative 
roots  of  the  proposed  equation.  Superior  and  inferior  limits 
to  the  positive  roots  of  the  transformed  will,  taken  negatively, 
be  inferior  and  superior  limits  to  the  negative,  roots  of  the 
proposed  equation. 

Ex.  5.     7^  —  13^2  +  38z  —  43  =  0. 

Merely  taking  the  coefficients  with  alternate  sign  changed, 
we  have     7+0  —  13  —  38  —  43,    from  which  we   obtain 

—  -f  1,  or  8,  as  a  superior  limit.     Hence  —  8  is  an  inferior 

limit  to  the  negative  roots  of  the  proposed. 

79.  The  limits  obtained  by  the  above  methods  are  often 
far  from  close.  Thus,  in  the  preceding  example,  8  was  ob- 
tained as  a  superior  limit  to  the  positive  roots  of 

7^4  _  13y2  _  SSy  _  43  _  0? 

while  3  is,  in  reality,  a  superior  limit.  Among  other  methods 
that  have  been  proposed,  that  of  Newton  gives  the  closest 
limit.  As  this  method  is,  however,  virtually  comprised  in  the 
method  of  analysis  given  in  (187),  we  refer  to  that  article. 

EXERCISES. 

By  one  of  the  preceding  methods,  find  superior  limits  to  the 
positive,  and  inferior  limits  to  the  negative  roots  of  the  follow- 
ing equations : 

1.  x*  +  15xL+  36x  —  54  =  0. 

2.  5xi  —  36a?  +  lSx  —  91  =  0. 

3.  2x5  —  W  +  9a3  —  llz2  +  13x  —  15  =  0. 

4.  x«  +  13.T3  —  39z2  —  54z  +  105  =  0. 


54  ALGEBRAICAL    EQUATION'S. 

80.  Prop.  IV.  —  If  two  numbers,  successively  substituted 
for  the  unknown  quantity  in  an  equation,  give  results  ivith 
contrary  signs,  these  numbers  include  between  them  an  odd 
number  of  the  roots  of  the  equation;  if  the  numbers  give 
results  tvith  the  same  sign,  they  include  either  an  even  number 
of  the  roots,  or  no  root. 

Let  p  and  q  be  respectively  superior  and  inferior  limits 
to  certain  real  roots  ax ,  a2,  as ,  .  .  .  .  am ,  of  an  equation 
f(x)  =  0. 

f(x)  =  (x—a{)  (x—a2)  (x—aB) ....  (x—am)  <p(x),      [1] 

where  <f>(x)  is  a  function  formed  from  the  product  of  quadratic 
factors  corresponding  to  imaginary  roots,  and  binomial  factors 
containing  roots  that  lie  without  the  limits  p  and  q,  so  that 
q>(x)  cannot  change  sign  for  any  value  of  x  between  p  and  q. 
If  in  [1]  we  put  p  and  q  successively  for  x,  we  have 

fiP)  =  (l>—<h)  {p—<h)  (p—as)..  ..(p-am)(f>(p), 
/(flO  =  (q-aj  (q-a2)  (q-ao). . .  .(q-am)<t>(q). 

Since  all  the  factors  (p—a^,  (p—a?), (p—am),  are  pos- 
itive, and  all  the  factors  (q—ai),  (q—a2), (q—am),  negative, 

while  (p(p)  and  <p(q)  have  the  same  sign,  f(p)  and  f(q)  will 
have  the  same,  or  contrary  signs,  according  as  the  number  of 

the  roots  aY,  a2,  as, am,  is  even  or  odd.    Hence,  if  p  and 

q  include  between  them  an  even  number  of  roots,  they  pro- 
duce results  with  the  same  sign,  if  an  odd  number,  results  with 
contrary  signs. 

Ex.      /  (x)  =  Xs  —  18z2  +  IOIjp  —  180  =  0, 
/(0)   =   -180, 

/(6)   =   -6, 
/(10)  =  30. 

As  /(0)  and  /(6)  have  the  same  sign,  we  infer  that  there  is 
either  no  root,  or  an  even  number  of  roots  between  0  and  6  ; 
there  are  really  two.  As  f(6)  and  /(10)  have  contrary  signs, 
we  infer  the  presence  of  some  odd  number  of  roots  between 
G  and  10. 


LIMITS    OF  THE    ROOTS    OF    EQUATIONS.  55 

81.  It  is  obvious  from  this  proposition  that  if  we  knew  a 
quantity  k  smaller  than  the  least  difference  between  any  two 
unequal  roots,  we  could,  by  substituting  in  succession  k,  2k,  3k, 
&c,  for  x,  obtain  as  many  changes  of  sign  as  there  are  unequal 
roots  in  the  equation,  and  thus  ascertain  their  number  and 
situation.  For  when,  in  the  course  of  our  substitutions,  we 
should  pass  the  least  root,  a  change  of  sign  would  apprise  us 
of  the  fact,  and  so  on  till  we  had  passed  all  the  unequal  roots. 
In  order  to  find  such  a  quantity,  less  than  the  least  difference 
of  the  roots,  Waring  proposed  to  transform  the  equation  into 
another  whose  roots  should  be  the  squares  of  the  differences  of 
the  roots.  Then,  supposing  we  found  /  as  an  inferior  limit  to  the 
positive  roots  of  the  transformed  equation,  VI  would  be  a 
quantity  less  than  the  least  difference  of  the  roots  of  the  pro- 
posed equation,  and  we  could  employ  it  as  above  suggested. 
This  method  of  separating  the  roots  is,  no  doubt,  theoretically 
perfect,  but  the  great  labor  of  forming  the  Equation  of  the 
Differences  for  equations  above  the  fourth  degree,  has  caused 
the  method  to  be  entirely  abandoned. 

82.  Peop.  V.  —  If  the  first  derived  function  f(x),  of  an 
equation  f{x)  =  0,  be  equated  to  zero,  a  real  root  of  f  (x)  =  0 
lies  between  every  adjacent  tivo  real  roots  of  f(x)  =  0. 

Let  «L ,  a2,  a3, a,n ,  denote  the  real  roots  of  /  (x)  =  0, 

arranged  in  descending  order  of  magnitude,  then 

f(x)  =  (x—ax)  (x—ov)  (x—ck) ....  (x-am)  0(jr)?      [1] 

where  0  (x)  is  a  function  that  cannot  change  sign,  being  the 
product  of  the  quadratic  factors  corresponding  to  the  imagi- 
nary roots,  if  any,  of  /  (x)  =  0. 

In  [  1  ]  we  put  y  +  z  for  x ;  then,  as  y  +  z  —  a 
—  z  +  y  —  a, 

f(y  +  z)  =  (z±y-al)(z  +  ^J(z  +  ^i3)...(z  +  ^m)<f)(y+z). 

If  now  the  first  member  of  this  identity  be  expanded  by 
Art.  7,  and  the  second  member  by  Art.  45,  in  ascending 
powers  of  z,  we  have 


56  ALGEBRAICAL  EQUATIONS. 

f(y)  +/i  (?)*  +  &c.  =  {Sm+Sm-1z+  . . .)  (0(y)  +  0i(3O*+  •  •), 

where  #„   =  the  product  of  all  the  terms  y-ax ,  y-a2,.  ^y-am  • 
and     S,,^  =  the  sum  of  the  products  of  the  same  terms,  taken 
m  —  1  at  a  time. 

Equating  the  coefficients  of  z,  we  have 

My)  =  &-10W  +  &01&). 

Substituting  for  #„  and  Sm.i  their  values,  and  putting  x  for 
?/,  as  it  is  now  immaterial  what  symbol  we  employ  in  this  last 
result,  we  have 

Mx)  =  \(x—(h)(oc-a3)...(x—am)^(x—al)(x--a3)...(x--am) 
+  ... }  0  (x)  +  (x—al)(x—a2)(x—a3) . . .  (x—am)  0X  (»). 

In  this  result  we  see :  (1),  Whichever  of  the  quantities 
ax ,  «2  ?  •  •  •  «m  ?  we  put  for  a,  the  coefficient  of  0X  (#)  will 
vanish ;  (2),  in  the  coefficient  of  0  (x),  when  we  put  for  x 
one  of  these  quantities  as  ari  all  the  products  will  vanish 
except  that  in  which  the  factor  x  —  ar  is  not  found ;  (3), 
this  product  will  determine  the  sign  of  f(ar),  as  0(#)  is 
always  positive.     Thus 

fx  (ax)   =  (ffi  —  a2)  (fl]  —  flg) (dfi  —  am)  0  («i), 

/l  (flz)     =     («2  —  «l)  («2  —  «3) («2  —  «m)  0  («2>> 

/i  (a3)  =  («s  —  fli)  (%  —  a2) (a*  —  am)  0  (a3), 

/i  («J  =  («»—  «i)  (am—  «2) (««—  0m-i)  0  (««)• 

These  products  are  alternately  positive  and  negative ;  for 
the  first  contains  no  negative  factor,  the  second  contains  one, 
(a2  —  flj),  the  third  contains  two,  (a$  —  ax)  and  (%  —  a2),  the 
fourth  contains  three,  and  so  on.  Hence,  by  the  preceding 
proposition,  an  odd  number  of  the  real  roots  of  fx  (x)  =  0 
must  lie  between  every  adjacent  two  of  those  of  f(x)  =  0. 

83.  In  the  preceding  article  it  has  been  assumed  that  the 
roots  8l3  #2,  az....am,  are  all  unequal.  The  conclusions 
there  arrived  at  hold,  however  small  the  differences  between 


LIMITS    OF   THE    ROOTS    OF    EQUATIONS.  51 

certain  of  the  roots  may  be.  Equal  roots  may  be  regarded  as 
roots  with  a  difference  infinitely  small,  and  the  proposition 
holds  with  regard  to  them  also,  as  may  be  seen  from  the  fol- 
lowing articles. 

84.  Cor.  1. — If  f{x)  =  0  have  r  roots,  each  equal  to  ax, 
then  f  (x)  ==  0  has  r  —  1  roots,  each  equal  to  ax . 

For  if  /  (x)  =  0  has  two  roots  each  equal  to  ax ,  then  the 
factor  x  —  ax  must  occur  in  each  of  the  products  that  form 
the  coefficient  of  (f>(x),  and  consequently  fi(a{)  =  0.  Thus, 
when  f(x)  has  two  factors  x  —  ai,f  (x)  has  one  such  factor. 
If  f(x)  has  three  factors  x  —  ax ,  f  (x)  has  two  such  factors  ; 
for  after  dividing  each  of  these  functions  by  x  —  a\ ,  the  quo- 
tient from  f(x)  will  still  have  two  factors  x  —  ax ,  and  there- 
fore the  quotient  from  f  (x)  must  have  one  such  factor. 
Thus,  generally,  when  fix)  has  r  factors  x  —  ax ,  f  (x)  has 
r  —  1  such  factors.  That  is,  when  f(x)  =  0  has  r  equal 
roots,  fi  (x)  =  0  has  r  —  1  equal  roots,  which  we  may  con- 
ceive as  lying  one  between  each  adjacent  two  of  the  r  equal 
roots  of  f(x)  =  0,  being  indeed  an  arithmetical  mean  be- 
tween these  two. 

The  same  observations  apply  to  any  other  groups  of  equal 
roots  in  f(x)  =  0.  Suppose  that  f(x)  =  0  has  the  root  a 
repeated  r  times,  the  root  b  repeated  s  times,  and  the  root  c 
repeated  t  times,  then  f(x)  and  f  (x)  have  (x— a)r~\x— b)s-x 
(x—cY~l  as  a  common  divisor. 

85.  Cor.  2. —  Only  one  root  of  the  equation  f(x)  =  0  can 
lie  between  any  adjacent  two  of  the  roots  of  f(x)  =  0.  For, 
if  there  could  be  two,  there  would  be  at  least  one  root  of 
f  (x)  =  0  lying  between  them,  so  that  the  roots  of  f  (x)  =  0, 
supposed  to  be  adjacent,  would  not  be  adjacent. 

86.  Cor.  3,— If  f(x)  =  0  has  m  real  roots,  then  f(x)  =  0 
must  have  at  least  m  —  1  real  roots  in  order  to  have  one  lying 
between  each  adjacent  two  of  those  of  f(x)  =  0.  Hence,  if 
an  equation  have  all  its  roots  real,  the  derived  equation  will 
also  have  all  its  roots  real,  each  lying  singly  between  a  pair 


58  ALGEBRAICAL    EQUATIONS. 

of  those  of  the  proposed  equation ;  the  derived  equation 
f  (x)  =  0  is  in  this  case  properly  called  the  limiting  equation 
of  f{x)  =0. 

87.  Since  f2  (%)  is,  Art.  4,  the  first  derived  function  of 
f\  (z)s  f-2  (X)  =  0  will  have  an  odd  number  of  its  roots 
lying  between  each  adjacent  pan  of  the  real  roots  of  f(x)  =  0. 
If,  then,  /  (x)  =  0  has  m  real  roots,  f  (x)  =  0  has  at  least 
m  —  1  real  roots,  and  f2  (x)  =  0  has  at  least  m  —  2  real 
roots.  Proceeding  in  this  way,  we  arrive  at  the  general  state- 
ment that,  if  f(x)  =  0  has  m  real  roots,  fr(x)  =  0  has 
at  least  m  —  r  real  roots. 

88.  Cor.  4.  —  If  /  (x)  =  0,  being  of  the  nth  degree,  has 
p  imaginary  roots,  it  has  n  —  p  real  roots ;  and  since,  by 
the  preceding  article,  f  (:r)  =0  has  at  least  n  —  p  —  r 
real  roots,  it  can,  being  of  the  (n  —  r)th  degree,  have  only  p 
imaginary  roots  at  most.  Hence,  if  any  of  its  derived  equa- 
tions have  p  imaginary  roots,  f(x)  =  0  must  have  at  least 
as  many. 

89.  Cor.  5.  — If  we  know  all  the  real  roots  of  f  (x)  =  0, 
we  can,  by  means  of  them,  ascertain  how  many  real  roots 
f(x)  =  0  contains. 

Let  a,  (3,  y,  .  .  .  .  k  be  the  real  roots  of  f  (x)  =  0,  ar- 
ranged in  descending  order  of  magnitude.  We  substitute 
these  quantities,  in  order,  for  x  in  f{x),  and  note  the  signs 
of  the  results,  /(a),  /(0),  /(y), /(«). 

Then,  according  as  /  (a)  is  negative  or  positive,  the  pro- 
posed equation  has,  or  has  not,  a  root  greater  than  a.  The 
equation  has,  or  has  not,  a  root  between  a  and  (3,  according 
as  f(a)  and  /(|3)  differ  or  agree  in  sign,  and  so  on.  Finally, 
if  /(«;)  be  positive  for  an  equation  of  odd  degree,  or  nega- 
tive for  one  of  even  degree,  there  is  a  root  of  f  (x)  =  0 
less  than  k,  otherwise  not.  The  number  of  the  real  roots 
of  f(x)  =  0  is  accordingly  equal  to  the  number  of  changes 
of  sign  in  the  series  of  results  produced  by  the  substitu- 
tions, in  order,  of  -f-  en,  a,  B,  y,  ....«,—  cc  for  x  in 
f(x)  =0. 


V 


LIMITS    OF    THE    ROOTS    OF    EQUATIONS.  59 

00.  This  property  of  the  roots  of  derived  equations  is 
the  basis  of  a  method  suggested  by  Rolle  for  separating  the 
real  roots  of  an  equation.  He  proposed,  by  means  of  the 
derived  equation  of  the  second  degree,  to  find  limits  to 
the  roots  of  the  preceding  derived  equation  of  the  third  de- 
gree ;  thence  limits  to  the  roots  of  the  antecedent  derived 
equation  of  the  fourth  degree,  and  so  on  till  limits  were 
obtained  to  the  roots  of  the  proposed  equation.  This,  called 
the  Method  of  Cascades,  has,  like  that  of  Waring,  been 
entirely  abandoned  on  account  of  the  length  of  the  calcu- 
lations required. 

Note. — An  equation  of  the  third  degree  can,  of  course,  be  easily  ana- 
lyzed in  this  manner,  since  its  first  derived  function  is  of  the  second  degree. 
It  appears,  however,  to  have  been  hitherto  overlooked  that  an  equation  of 
the  fourth  degree  can  also  be  analyzed  by  the  aid  of  a  quadratic,  or  that, 
generally,  any  equation  in  which  the  second  term  from  beginning  or  end  is 
absent  (a  condition  that  can  always  be  fulfilled  by  Art.  71),  can  be  analyzed 
by  the  aid  of  an  equation  lower  by  two  degrees.  Let  the  equation  be, 
Cnxn  +  0  +  GVstf'1-2  +  .  .  .  .  C-2x*  +  &x  +  0.  =  0.        [1]. 

Forming  (64)  the  equation  of  the  reciprocals  of  this,  and  equating  its 
first  derived  function  to  zero,  we  have 

nCoy~l  +  {n-l)GitT^  +  ..,..  3<7„_32/2  +  2C„_22/  =  0.      [2]. 
An  odd  number  of  the  roots  of  [2]  lies  between  every  adjacent  two  of 
the  reciprocals  of  the  real  roots  of  [1]  ;  therefore,  taking  the  equation 

2Cn_2z»-1  +  3&-3S*-2  +  .  .  .  .  (/z-l)Ciz2  +  jiCoZ  =  0,  [3] 
whose  roots  are  the  reciprocals  of  those  of  [2],  we  have  an  equation  whose 
roots  separate  those  of  [1].  One  root  of  [3j  is  z  =  0 ;  the  remaining  roots 
can  be  determined  by  an  equation  of  the  degree  n  —  2.  If  we  suppose 
[1]  to  be  of  the  fifth  degree,  its  roots  can  be  separated  by  those  of  the 
cubic  2C3s3  +  3C2  s2  4-  4G'i  z  +  5Co  =  0,  and  another  root  2  =  0.  If  the 
equation  be  of  the  fourth  degree,  its  roots  will  be  separated  by  those  of 
the  quadratic  2C2z2  +  W\Z  +  4 Co  =  0,  which  are  given  in  the  formula 
e  _  -sd±  VW-zw,c0^  and  another  root  g  _  0 

Ex.  1.  The  equation  a4— 12a?  +  12a— 3  =  0  has  for  roots  x  =  -3.9.  ., 
x  =  .44. .,  .60. .,  2.8. .  ;  by  the  above  formula  we  obtain  z  =  0,  .5,  1, 
which  separate  the  values  of  x. 

Ex.  2.  The  equation  a4  -  467a;2  +  3660a;  +*2826  =  0  has  for  roots 
x  =  12.70828. .,  12.70820. .,  -.70. .,  -24.7. . ;  by  the  aid  of  the  formula 
we  obtain  z  =  12.70824. .,  0,  —  .95.  .,  which  separate  the  roots  of  the 
equation,  two  of  which  concur  to  six  figures.  We  thus  see  that  equa- 
tions of  less  than  the  fifth  degree  can  be  easily  analyzed  by  means  of 
limiting  equations. 

Though  an  equation  of  the  third  degree  in  its  most  general  form  may 
be  analyzed  by  means  of  a  quadratic,  it  may  be  of  use  to  note  that  when 
in  the  form  C?  xz  +  C\  x  +  Co  —  0,  its  two  smaller  roots,  which  are  of  the 

same  sign  as  Co,  can  be  separated  by  3 Co  -. 2Ci,  if  they  are  real.     For 

example,  the  equation  xs— 7x  +  7  =  0  has  one  root  greater  than   ~  =  1.5, 

and  one  less.     For  exercises  the  student  is  referred  to  the  numerous  ex- 
amples given  on  pages  125  and  153. 


60  ALGEBRAICAL  EQUATIONS. 


CHAPTEE    VI. 

ON    THE    DEPRESSION    OF    EQUATIONS. 

91.  In  the  present  chapter  we  shall  treat  of  those  equations 
which,  on  account  of  certain  relations  existiDg  among  the 
roots,  are  capable  of  depression,  that  is,  of  being  resolved  into 
equations  of  lower  degrees. 

Among  the  most  important  of  these,  are  those  equations 
which  contain  equal  roots.  We  propose  to  show  how  these 
equal  roots  may  be  eliminated,  and  the  solution  of  the  equa- 
tion containing  them  reduced  to  that  of  equations  of  lower 
degrees  having  only  unequal  roots. 

92.  Prop.  I.  —  An  equation  f(x)  =  0  has,  or  has  not, 
equal  roots  according  as  f '  (x)  and  f\(x)  have,  or  have  not, 
a  common  factor  involving  x. 

As  far  as  regards  real  roots,  this  has  been  shown  in  Art.  84, 
and  that  result  may  be  made  to  include  imaginary  roots  by  a 
slight  modification  of  the  demonstration. 

Thus,  let  «! ,  a2,  (t3 ,  .  .  .  .  an  include  all  the  roots  of 
/  (x)  =  0,  real  or  imaginary ;  then 

f(x)  =  (x—al)(x—a2)(x—a^  ....  (x— a,),  [1]. 

fx(x)  =  \(x—a2)(x—a<i)...(x—all)  +  (x—al)(x—a;i)...(x—a)) 

+  &c    \     [2]. 

It  is  evident  that  f(x)  and  fx{x)  have  no  common  factor 
if  all  the  factors  in  f (x)  are  different;  for  then  all  the 
products  in  [2]  are  different,  each  being  equal  to  f{x)  divided 
by  a  factor  (x— av),  (x—a?),  &c,  different  in  each  case  ;  or, 

m  =  m  +  m'+m  +  ....m. . 

1    w         x— a{         x—a1         x—a3  x—an 


DEPRESSION"    OF    EQUATIONS.  61 

If  in  [1]  we  suppose  r  of  the  roots  to  be  each  equal  to  a, 
s  of  them  equal  to  b,  and  t  of  them  equal  to  c,  then 

fx(x)  =  \{x-ay-\x-by(x-cy .  ..(X-an)  +  {x-ay{x-l)-' 

(x—c)1. . .  (x—an)  -j-  {x—a)r(x—b)8{x—c)t~i. . .  (x—a>t) 
+  products  each  containing  (x— a)'(x— by(x— c)1  }. 

Thus  (x — a)'~1(x—by~1(x — c)<_1  occurs  as  a  factor  in  every 
term  of  fx  (x),  and  is  therefore  a  common  factor  of  /  (x) 
and  /i  (x). 

93.  In  order,  therefore,  to  determine  whether  an  equation 
f{x)  =  0  has  equal  roots,  Ave  have  only  to  ascertain  whether 
f(x)  and  fi  (x)  have  aAcommon  divisor  </>  (a).  If  such  a  com- 
mon measure  be  found,  then  the  quotient  of  f(x)  by  <j>  (x) 
will,  equated  to  zero,  contain  all  the  roots  of  the  proposed, 
without  repetition.  If  0  (x)  is  of  the  form  (x— a)r,  it  will  be 
found  advantageous  to  divide  by  (x— a)r+\  so  as  to  obtain  the 
quotient  of  as  low  degree  as  possible. 

Ex.  1.  Given  f(x)  =  a4  —  a3  —  30a;2  +  32a;  +  160  =  0 ; 
required  to  determine  whether  the  equation  has  equal  roots. 

Here  fY  (x)  =  4a3  —  3a2  —  60a  +  32,  and  we  find  a  com- 
mon measure  (x— 4) ;  /(a),  therefore,  has  a  factor  (a— 4)2, 
dividing  by  which,  we  find 

f(x)  =  (£-4)2  (z2+  7a+10)  =  0. 

Thus  the' roots  of  the  equation  are,   —5,  —2,  4,  4. 

Ex.  2.  Given  f(x)  =  4a5  +  17a4  +  8a3  —  40a;2  —  32a  +  16 
=  0  ;   determine  whether  the  equation  has  equal  roots. 

Here  fx  (x)  =  20a*  +  68a3  +  24a2  _  80a  —  32,  and  we 
find  a  common  measure  a2  +  4a  -f-  4 ;  /  (a)  has,  therefore, 
a  factor  (a  +  2)3.     Dividing  by  this,  we  find 

f{x)  =  (a  +  2)3  (4a2-  7a  +  2)  =  0. 

Thus  the  roots  of  the  equation  are,  —  2,  —  2,  —  2,  and 

i(7±VT7). 

94.  The  common  measure  of  /(a)  and  fY  (x)  may,  how- 
ever, be  itself  an  expression  containing  more  than  one  set  of 


62  ALGEBRAICAL    EQUATIONS. 

repeated  factors.  We  require,  therefore,  to  deduce  a  systematic 
process  for  obtaining  separate  equations  that  contain  only  one 
set  of  factors. 

Let  Xi  be  the  product  of  all  the  factors  that  occur  but 
once  in  f(x) ;  X2 ,  the  product  of  the  factors  that  occur 
twice ;  X3 ,  the  product  of  those  that  occur  three  times ; 
and  so  on,  any  of  these  factors,  as  Xm ,  being  unity  if  there 
is  no  set  of  factors  repeated  m  times.  Thus  f(x)  =  Xt 
X2*   Xi  X4*....X/. 

Denoting  the  G.  C.  M.  of  f(x)  and  f1  (x)  by  (b{  (x),  we 
have, 

^(x)  =  XtX£X}::..xr\ 

and  denoting  the  G.  C.  M.  of  </>!  (x)  and  its  derived  function 
(pi'(x)  by  02  (x),  we  have, 

02  (x)  =  Xz  X? X;-1. 

In  like  manner  we  obtain  in  succession, 

03  (x)  =  X4X52 X/-3, 

04  (x)  =      xa  . . . .  x;-\ 

0,-1  (x)   =  Xr. 

0,.   (x)   =  1,  since  Xr  has  only  unequal  factors. 

From  these  functions  we  obtain  by  division, 

*  M  =  $|=*M.."*, 
Fr  (x)  =*£&=  xr. 

<p,.(x) 

We  can  now  obtain  the  separate  factors  Xt ,  X2 , X,_] , 

by  another  division  ;  thus, 

Fl    (*')    _     Y    .       F2(x)    _  Fr-Ax)    _     v 

W(xj  -  Al '  TO  ~Xi' ~fM  ~  X-1- 


DEPRESSION    OF    EQUATIONS. 

Finally,  by  equating  the  functions  A\  .  X: X  .  to  zero. 

we  obtain  the  roots  of  the  proposed  equation  :   any  root  that 
occurs  in  Xm ,  for  example,  occurring  m  times  in  f(x)  =  0. 
The  process  may  be  presented  as  follows  : 

/  (x),  o^x),  <fo(x),  ....  </>,_!  (X), 
Fx{x),  Fax),  F3{x),  .  .  .  .  Fr  (x), 
Ji\      ,      A 2      ,      X3      ,  .    .    .    .  Ar        , 

In  the  first  line,  each  term,  after  the  first,  is  the  G.  C.  M. 
of  the  preceding  term  and  its  first  derived  function.  In  the 
second  line,  each  term  is  the  quotient  of  the  term  under 
which  it  stands,  by  the  next  term  in  the  same  line.  In  the 
third  line,  each  term  is  the  quotient  of  the  term  under  which 
it  stands  by  the  next  term  in  the  second  line. 

The  following  example  will  illustrate  the  process. 

/  (x)  =  Xs  —  ~x7  —  2z*  +  IIS.?5  —  2o9x*  -  83a3  +  613a8 

-  lOSz  -  432  =  0. 
<h  (x)  =  x*  —  W  +  13a*  +  ox  —  18. 

02  (x)  —  x  —  3. 

03  (X)    =   1. 

Fx(x)  =  x*  —  ld.r3  -f  102-  +  'U. 

•  F2(x)  =  x5  —  4a*  +  x  -f-  6. 

F;,{x)  =  x-3. 

X,  =x  +  ±. 

X,  =  x2  —  x  —  2. 

XZ  =  x  -  3. 

.-.     /  (x)  =  (s+4)  (x^-x-2Y(x-S)K 

95.  Cor.— When  the  coefficients  of  f(x)  are  all  commen- 
surable, the  functions  Xi ,  X2 , Xt .  have  likewise  all  their 

coefficients  commensurable.     Hence 

(1).  If  only  one  of  the  roots  of  f(x)  =  0  is  repeated  r 
times,  that  root  must  be  commensurable ;  for  it  will  be  de- 
termined by  an  equation  Xr  =  0,  which  will  be  of  the  first 
degree,  and  contains  no  incommensurable  quanti 

(2).  Hence  incommensurable  equal  roots  will  be  deter- 
mined by  equations  of  at  least  the  second  degree  :  and  any 
equation  that  contains  such  roots  must  have  a  factor  X    . 


<J4  ALGEBRAICAL    EQUATIONS. 

where  m  cannot  be  less  than  two;  that  is,  the  equation  must, 
at  least,  have  a  factor  X?  of  the  fourth  degree.     Hence 

(3).  An  equation  of  the  third  degree  with  commensurable 
coefficients  cannot  have  equal  roots  that  are  not  commen- 
surable ;  and 

(4).  An  equation  of  the  fourth  degree  with  commensurable 
coefficients  cannot  have  incommensurable  equal  roots,  unless 
its  first  member  is  a  perfect  square ; 

(5).  An  equation  of  the  fifth  degree  cannot  have  incommen- 
surable equal  roots,  unless  it  has  one  commensurable  root. 

(6).  Equations  of  the  sixth  and  higher  degrees  may  have 
such  roots,  even  when  their  coefficients  are  all  commensurable. 

96.  We  are  thus  in  possession  of  a  method  by  which  we 
can  eliminate  all  equal  roots  from  an  equation,  and  obtain 
other  equations  of  inferior  degrees  involving  roots  without 
any  repetition.  The  process,  however,  though  simple  in 
theory,  involves  much  numerical  labor  in  equations  of  high 
degree,  with  large  coefficients,  as  will  be  fully  appreciated 
when  we  come  to  exemplify  Sturm's  Method,  which  consists 
essentially  in  an  operation  similar  to  that  of  finding  the 
G.  0.  M.  of  a  proposed  function,  and  its  first  derived  function. 

EXERCISES. 

Find  the  equal  roots  in  the  following  equations  : 

1.  ^3  _  X2  _  $x  -f.  12   =r   0. 

2.  4i3  +  12x2  —  63#  +  54  =  0. 

3.  as*  —  W  +220a;  +  75  =  0. 

4.  8xi  —  52£3  +  G6x2  +  llx  —  49  =  0. 

5.  x5  +  xi  —  l&c3  +  14a:2  +  21x  +  5  =  0. 

6.  x6  -  13z4  +  12^3  +  9W  +  QGx  +  12  =  0. 

RECIPROCAL    EQUATIONS. 

97.  In  Art.  64  we  found  that  an  equation 

xn  4  Cn.xxn-1  +  <7,(_2 £"-2 -}- C.x'1  4-  dx  4-  C0  =  0 

may  be  transformed  into  an  equation  whose  roots   are  the 


DEPRESSION    OF    EQUATIONS.  G5 

reciprocals  of  those  of  the  proposed,  by  simply  writing  the 
coefficients  in  reverse  order,  and  replacing  x  by  some  other 
symbol.  If  the  resulting  transformed  equation  be  identical 
with  the  proposed  equation,  it  is  obvious  that  for  every  value  a 

that  satisfies  either  equation,  there  must  be  another  value  - 

that  also  satisfies  it.  Such  equations  are  called  reciprocal 
equations,  and  it  will  be  shown  that  they  can  be  depressed  to 
equations  of  one-half  their  own  degree. 

98.  To  find  what  relations  must  exist  among  the  coeffi- 
cients in  order  that  a  proposed  equation  may  be  a  reciprocal 
equation,  suppose  the  equation  above  to  be  transformed  by  the 
rule  in  (64),  and  divide  through  by  C0 ;  thus  we  obtain 

r  +  gy-1  +  §V-2+ . . .  .%v  +  %*  + 1  =  o. 

Iii  order  that  the  transformed  may  be  identical  with  the 
proposed  equation  we  must  have 

C 

77  —  k/i-i  >  or  Ci  =  C  o^n-i  j  ^2  —  G0G„_2 ; . . .  Cr  =  Co6„_r ;  1  =  G0~. 

From  the  last  equation  we  obtain  C0  =  +1,  or  C0=  —1, 
and  this  gives  rise  to  two  classes  of  reciprocal  equations. 

First.    If  C0  =  +1,  we  have 

d  =  Cn_, ;     C2  =  Cn_2 ; . . . .  Cr  =  C/(_, ; . . . .  &c. 

Therefore  «w  equation  is  a  reciprocal  equation  when  co- 
efficients equidistant  from  the  first  and  last  are  equal. 

Secondly.     If  C0  =  —1,  we  have 

L>\  =  —  6„_! ;     62  =  —  Cn_2 ; . . . .  Cr  =  —  G„_,. ;  &c. 

In  this  case,  supposing  n  =  2m,  we  should  have  the  middle 
term  Cm  =  —  Cm ,  which  is  impossible  unless  Cm  =  0. 

Therefore  an  equation  is  a  reciprocal  equation  when  co- 
efficients equidistant  from  the  first  and  last  are  numerically 


G6  ALGEBRAICAL  EQUATIONS. 

equal  though  of  contrary  signs,  provided  that  the  middle  co- 
efficient he  zero,  if  the  equation  be  of  even  degree. 

Ex.  1.  xG  —  Hx5  -f  45s4  +  12z3  +  45z2  —  lx  +  1  =  0,  is 
a  reciprocal  equation  of  the  first  class. 

Ex.  2.  x8  +  llz7  —  5^  —  Six5  +  Six?  +  5x2 -lk-l  =  0, 
is  a  reciprocal  equation  of  the  second  class. 

99.  A  reciprocal  equation  of  the  first  class,  and  of  odd 
degree,  is  obviously  satisfied  by  x  =  —  1,  and  has,  therefore, 
its  first  member  divisible  by  x  +  1.  Since  the  remaining 
roots  occur  in  reciprocal  pairs,  the  quotient  arising  from  this 
division  will,  equated  to  zero,  be  a  reciprocal  equation  of  even 
degree  with  its  final  term  positive. 

In  like  manner,  a  reciprocal  equation  of  the  second  class, 
and  of  odd  degree,  has  its  first  member  divisible  by  x  —  1, 
and  can  thus  be  depressed  to  a  reciprocal  equation  of  even 
degree  with  its  final  term  positive. 

A  reciprocal  equation  of  the  second  class,  and  of  even  degree, 
is  evidently  satisfied  by  both  x  =  1,  and  x  =  — 1 ;  its  first 
member  is  therefore  divisible  by  x2  —  1,  and  will  furnish  a 
quotient  which,  equated  to  zero,  is  a  reciprocal  equation  of 
even  degree  with  its  final  sign  positive. 

Every  reciprocal  equation  is  therefore  either  of  even  degree 
wTith  its  final  sign  positive,  or  may  be  reduced  to  that  form 
preparatory  to  depression  by  the  following  theorem 

100.  Prop.  I. — A  reciprocal  equation  of  even  degree  with 
its  last  term  positive,  may  he  depressed  to  an  equation  of  one- 
half  the  degree  of  the  proposed. 

Let    x-m  +  Cx  x*m~l  +  C2  x^-2  +  . . .  C2  x2  +  Cx  x  + 1  =  0      [1] 
be  the  proposed  equation. 

Collecting  in  pairs  the  terms  that  are  equidistant  from  the 
first  and  last,  and  dividing  by  xm,  we  have 

(z'l'  +  ^)+Ci(z-"-'  +  ^  +  cJyx^-  +  ^)  +  &c.=  0.     [2]. 


DEPRESSION    OF    EQUATIONS.  G7 

Let 
x  H =  y  ;     then 

*  +  J  =  f  -  2> 

(■'"" + h) = (x""1 + ^)  (* + 9  ~  (** + sM = r  ~  mr~2 + &0, 

By  substituting  these  values  in  [2] ,  we  obtain  an  equation 
of  the  m"'  degree  in  y,  that  is,  an  equation  of  half  the  degree 
of  the  proposed.    Any  root,  as  a,  of  this  equation  will  give 

two  roots  of  the  proposed  by  means  of  the  relation  x  +  -  =  a; 

X 

or,  x2  —  ax  +  1  =  0. 
Ex.     Let  2a;6  —  Ux5  +  19a4  —  19a;2  +  12a;  -2  =  0. 

Since  both  +  1  and  —  1  are  roots  by  inspection,  we  divide 
the  left  hand  member  by  x2  —  1,  and  obtain, 

2x*  —  12a;3  +  21a;2  —  12a;  +  2  =  0, 
therefore        2  (x2  +  -\  —  n(x  +-)  +  21  =  0, 

and  putting  x  -\ — -  =  ?/,  we  obtain 

x 

2(y2-2)  -Uy  +  21  =  0, 
or  2\f          —  12y  +  17  =  0, 

thence  y    =    J  (6  +  a/2), 

and  x    =     2  ±V2  oy  ^(2  ±  V%. 

EXERCISES. 

Solve  the  following  reciprocal  equations : 

1,  3a;4  -  7a;3  +  31a;2  -  7a;  +  3  =  0. 

2.  5a4  +  82"3  —  5Qx2  +  8a;  +  5  =  0„ 


68  ALGEBRAICAL  EQUATIONS. 

3.  a5  +  14a4  —  25a3  —  25a2  +  14a  +  1  =  0. 

4.  8a5  —  52a4  —79a3  +  79a2  +  52a  —  8  =  0. 

5.  a6  —  23a5  —  84a4  +  84a2  +  23a  —  1  =  0. 


BINOMIAL    EQUATIONS. 

101.  These  equations  are  such  as  consist  of  two  terms 

only,  as, 

xn  —  A   =  0, 

where  A  is  a  known  quantity.  This  is  the  only  extensive 
class  of  equations  that  are  capable  of  complete  solution  by  a 
general  method.  Their  general  solution,  however,  depends  on 
De  Moivre's  formula,  and  is  contained  in  works  on  Trigo- 
nometry. The  following  propositions  comprise  the  theory  of 
Binomial  Equations  and  the  solution  of  such  cases  as  are 
readily  solvable  by  algebraical  processes. 

102.  Prop.  I. — The  roots  of  a  binomial  equation  are  all 
different. 

For  the  first  derived  function  of  a"  —  A  is  nxn~l,  and  no 
value  of  x  can  make  a"  —A  and  nxn~l  vanish  simultaneously. 
See  Art.  92. 

103.  Cor.  —  Any  algebraical  quantity  (that  is,  any  quan- 
tity included  under  the  general  form  a  +  bV — i,  where 
a  and  b  are  real  quantities,  positive,  negative,  or  zero),  has 
n  different  nth  roots.  From  the  above  equation  we  have 
x  =  \/A,  and  a  has  n  different  values  (36). 

104.  Prop.  II. — All  the  nth  roots  of  an  algebraical  quan- 
tity may  be  found  by  multiplying  one  of  them  by  the  n  differ- 
ent nth  roots  of  unity. 

For  in  the  equation  a"  —  ^4  =  0  put  ay  for  a,  where  a  de- 
notes one  of  the  nth  roots  of  A.    Then  we  have 

anyn  ==  an;    or,  yn  =  1. 

From  this  we  have  y  =  yX  Hence  a  =  \  A  =  ay 
=  «#!. 


DEPRESSION    OF    EQUATIONS.  69 

105.  Since  we  can  always,  as  in  the  preceding  article,  re- 
duce a  binomial  equation  to  the  form  x"  ±  1  =  0,  we  shall, 
in  the  following  propositions,  confine  our  attention  to  this 
form. 

106.  Prop.  III.  —  If  a  be  any  root  of  the  equation 
xn  —  1  =  0,  then  any  integral  power  of  a  will  also  ~be  a  root. 

For  (a"')n  =  amn  =  (a")"1  =  1'"  =  1. 

107.  Cor.  —  It  hence  appears  that  the  roots  of  the  equa- 
tion xn  —  1  =  0  may  be  represented  under  an  infinite  variety 

of  forms,  since  each  term  of  the  series   ar™, a-3,   a-2,   arl, 

a0,  c,  a2,  a3, . . . .  is  a  root.  Of  these  there  cannot,  however, 
be  more  than  n  essentially  different,  as  otherwise  the  equation 
would  have  more  than  n  roots. 

108.  Prop.  IV.  — If  a  be  any  root  of  xn  -f  1  ==  0,  then 
any  odd  integral  power  of  a  will  also  be  a  root. 

For  (am)n  =  (an)m  =  (  —  l)m  =  —1,   if  m  is  odd. 

109.  Prop.  V.  —  If  m  be  prime  to  n,  the  equations 
xm  —  1  =  0,  and  xn  —  1  =  0,  have  no  common  root  but 
unity. 

Let  p  and  q  be  integers  such  that  pm  —  qn  =  1,*  and 
suppose  a  is  a  root  common  to  the  two  equations.  Then,  as 
am  =  1,  and  an  =  1,  we  have  also  a^"1  =  1,  and  a*1  =  1 
(105).  Hence,  by  division,  a*"'1-*'1  =  1,  or  a1  =  1 ;  that  is, 
the  only  common  root  is  unity. 

110.  Prop.  YI.~- If  n  is  a  prime  number,  and  a  an 
imaginary  root  of  xn  —  1  =  0,  then  all  the  roots  of  the  equa- 
tion are  found  in  the  series,  1,  o,  a2,  a3, a'1-1. 

For  these  are  all  roots  by  Prop.  V ;  and  no  two  of  them 
are  equal.  For,  if  possible,  let  a^  =  a«,  then  a^_«  =  1, 
that  is,  a  is  a  root  of  xp~q  —  1  =  0,  and  also  of  xn  —  1  =  0, 
which  is  impossible,  since  p  —  q  being  less  than  n  must  be 
prime  to  it. 

*  Such  integers  can  always  be  found  by  algebra. 


70  ALGEBRAICAL  EQUATIONS. 

111.  Prop.  VII.  —  The  solution  of  xn  —  1  =  0,  tvhere  n 
is  a  composite  number,  may  be  made  to  depend  upon  the  solu- 
tion of  the  equations  xp  —  1  =  0,  xq  —  1  =  0,  &c,  where 
p,  q,  r,  &c,  are  the  different  prime  factors  of  n. 

First  suppose  n  =  pq,  where  p  and  q  are  prime  to  each 
other.  Then  xn  —  1  =  0,  or  xpq  —  1  =  0,  is  divisible  by 
both  xp— 1  and  xq—l.     By  Prop.  VI,  the  roots  of  #p— 1  =  0, 

are, 

1,  a,  a2,  a3,  .  .  .  .  a*-1, 

and  those  of  a?  —  1  =  0,  let  us  suppose  are, 

1,  ft  02,  0», . . . .  0*-1, 

and  all  these  are  roots  of  xn  —  1  =  0.  Also  the  products 
formed  by  multiplying  each  term  in  the  first  row  by  each 
term  in  the  second  are  all  roots.  For  each  of  these  products 
is  of  the  form  ar(3\  and  since  a"1  =  1,  and  (3sn  =  1,  therefore 
(ar(3s)n  =  1,  that  is,  ar  (3s  is  a  root.  Moreover,  no  two  of  these 
products  are  alike.  For,  if  possible,  suppose  ar(3s  =  a*  (2°, 
then  ar~l  —  (3V~S.  But  as  ar~l  is  a  root  of  xp  —  1  =  0,  and 
ft-*  a  root  of  xq  —  1  =  0,  these  equations  have  a  common 
root  besides  unity,  which  (110)  is  impossible,  since  p  and  q 
are  prime  to  each  other.  Therefore  the  pq  products,  formed 
by  multiplying  each  root  of  xp  —  1  =  0  by  each  root  of 
xq  —  1  =  0,  are  the  roots  of  xpq  —  1  =  0,  or  x11  —  1  =  0. 

In  the  same  way,  if  n  be  the  product  of  three  prime  factors 
p,  q,  r,  it  may  be  proved  that  the  roots  of  xn  — 1  =  0  are 
the  pqr  products  obtained  by  multiplying  together  the  roots 
of  the  equations  xp  —  1  =  0,  &  —  1  =  0,  xr  —  1  =  0  ;  and 
similarly  for  any  number  of  prime  factors. 

112.  Again,  let  n  be  a  power  of  some  number,  as  n  =  p*. 

Suppose  the  roots  of  xp  —  1  =  0   are 

1,  a,  a2,  a3,  .  .  .  .  ap~\ 
then  these  as  well  as 

1,   ^a;   <V~a\   ya%....$ap-' 

are  roots  of  xp*—  1  =  0,  as  is  also  the  product  of  each  root 
in  the  first  row  by  each  root  in  the  second.    We  have  there- 


DEPRESSION    OF    EQUATIONS.  71 

fore  p2  different  quantities  of  the  general  form  ar  •  tya*  all 
satisfying  the  proposed  equation,  and  these  are  therefore  all 
the  roots. 

But  by  the  solution  of  xp  —  1  =  0  we  obtain  only  the  p 
roots  given  in  the  first  row  above;  those  in  the  second  rowT 
must  obviously  be  determined  by  solving  the  proposed  equa- 
tion. A  similar  remark  applies  when  n  =  jf,  where  r  =  3, 
or  any  greater  number. 

Tims  wTe  can  find  all  the  roots  of  x15  —  1  =  0  by  solving 
^-1  =  0,  and  x3  —  !  —  0;  but  if  Xs  —  1  =  0  be  given, 
we  can  find  by  this  method  only  five  roots,  and  if  x49  —  1  =  0 
be  given,  only  seven  roots. 

113.  Prop.  VIII. — The  solution  of  mi  equation  af  ±  1  =  0 
may  be  reduced  to  that  of  an  equation  cf  not  more  than  one- 
half  the  degree  of  the  projjosed  equation,  and  having  all  its 
roots  real. 

For,  taking  the  most  unfavorable  case,  xn  +  1  =  0,  where 
n  is  an  even  number,  and  the  equation  has  therefore  no  real 
root,  we  see  that  this  is  a  reciprocal  equation  of  even  degree 
with  its  final  term  positive;,  it  may  therefore  (100)  be  depressed 
to  an  equation  in   y  of  one-half  the  degree  of  xn  -f-  1  =  0. 

As  #  =.  x  +  - ,  or  x2  —  yx  +  1  =  0,  y,  being  the  sum  of 
a  pair  of  conjugate  roots,  is  real. 

X"  —  1  =  0,  when  n  is  even  may  be  depressed  two  degrees 
by  dividing  by  x2  —  1 ;  and  xn  ±1  =  0,  when  n  is  odd,  may 
be  depressed  one  degree  by  dividing  by  x  ±  1.  In  either  case 
the  depressed  equations  are  reciprocal  equations  of  even  degree 
with  their  final  terms  positive,  and  these  again  may  be  de- 
pressed to  equations  in  y  of  less  than  one-half  the  degree  of 
the  proposed  equations. 

Ex.  1.  z3  —  1  =  0, 

.-.     (x—l)(x2  +  x  +  l)  =  0. 

Hence  the  roots  are  1,  and  —  -J-  (1  ±  V—  3),  which  values 
are  the  cube  roots  of  unity.  By  changing  their  signs,  we 
obtain  the  three  cube  roots  of  —  1. 


72  ALGEBKAICAL    EQUATIONS. 

Ex.   2.  &*  -f  1    =   0. 

Putting    y  =  x  -\ — ,    we   obtain    y2  —  2  =  0,    whence 

y  =  ±V2, 
and      x*  +  1   =  (a3  +  a/2  ■  a+1)  (z2—  \/2  ■  s  +  1), 

from  which  we  obtain  as  the  four  fourth  roots  of  unity, 

i(V2±  V^2),   and    -  i(V%  ±  V^2). 

Ex.  3.  x5  —  1   =  0. 

From  this,  dividing  by  x  —  1,  we  obtain 

x±  +  #3  +  a2  +  x  +  1  =  0. 
Putting    ^  =  a;  +  - ,  we  have 

V2  +  y  —  1  =  0  j    whence   ?/  =  —  -J-  (1  ±  a/5). 
Therefore 
a:5_1  _  (g^l)  (g?  +  *  +  /5  •  x  +1)  (a?  +  1~»       -g+1), 

/y  /y 

whence  we  obtain  as  the  five  fifth  roots  of  unity, 


1,  i(A/5-l±j/-10-2v/o),  -1^5  +  1  ±j/- 10 +  2\/o). 
These,  with  changed  signs,  are  the  fifth  roots  of  —  1. 
Ex.  4.  xG  —  1  =  0. 

By  Prop.  V,  the  solution  will  be  effected  by  finding  the 
roots  of  a?  —  1  =  0,  which  are  1  and  —  J  (1  ±  V—  3),  and 
those  of  x2  —  1  =  0,  which  are  1  and  —  ]  ;  the  products 
of  these  sets  of  roots  will  furnish  the  six  sixth  roots  of  unity. 

114.  When  we  proceed  in  the  same  way  to  solve  x1— 1  =  0, 
we  obtain  a  reduced  equation  in  y  of  the  third  degree ;  for 
x9  —  1  =  0,  a  reduced  equation  in  y  of  the  fourth  degree, 
and  so  on. 

These  algebraical  methods  of  obtaining  the  roots  of  bi- 
nomial equations  are  of  little  practical  importance,  since,  as 
before  mentioned,  the  roots  of  such  equations  are  most  readily 


DEPRESSION    OF    EQUATIONS.  73 

and  symmetrically  obtained  by  the  aid  of  De  Moivre's  For- 
mula. 

115.  Besides  the  three  preceding  classes  of  equations,  any 
equation  is  capable  of  depression  when  we  know  that  the  roots, 
or  some  of  them,  are  known  functions  of  each  other.  If  the 
roots,  for  example,  are  known  to  be  in  geometrical,  or  arith- 
metical progression,  we  can  by  means  of  the  known  relations 
between  the  roots  and  coefficients,  Art.  45,  easily  depress  the 
proposed  equations  to  others  of  lower  degree. 

In  general,  if  the  relation  a2  =  (j>  («i)  is  known  to  exist 
among  two  of  the  roots,  ax,  and  a2,  of  f(x)  =  0,  we  may 
depress  the  equation  as  follows  : 

Substitute  </>  (x)  for  x  in  f(x),  and  let  F(x)  be  the  result- 
ing function.  Then  we  have  both  f(x)  =  0,  and  F(x)  =  0, 
when  x  =  ax .  For  aY  is  a  root  of  f(x)  =  0  by  supposition, 
and  to  put  ax  for  x  in  F{x)  is  equivalent  to  putting  a2  for  x 
in  f(x).  Therefore  f(x)  and  F(x)  have  a  common  factor 
x  —  aL  which  can  be  found.  Then  ax  being  known,  a2  =  0  («i) 
becomes  known,  and  the  equation  may  be  depressed  two  di- 
mensions. 

Ex.     Suppose  it  is  known  that  two  roots  aY  and  a2  of 

a*  _  9^3  +  7^2  +  6a;  +  16  =  0,  [1] 

are  such  that  a2  =  3«L  -f  2. 

Substitute   3x  -f  2  for  x  in  the  equation,  then 

(33  +  2)4  —  9(3^  +  2)3  +  7(3^  +  2)  +  6(3a;  +  2)  +  16  =  0, 
or,  81a*  —  2W  —  207a;2  —  126a;  +  0  =  0, 

or,  9a;3  —  3a;2  —  23a;  —  14  =  0.  [2]. 

We  find  (a;-2)  as  the  G.  C.  M.  of  [1]  and  [2].  Therefore 
cii  =  2,  and  a2  =  3ax  +2  =  8.     Thus  we  find 

34  _  9a-3  +  7a;2  +  63  +  16  =  (3-2)  (3—8)  (a;2  +  a;  +  l), 

and  the  depressed  equation  is 

x2  +  x  +  1  =  0. 


74  ALGEBRAICAL  EQUATIONS. 

116.  Whenever  in  the  course  of  the  analysis  of  an  equation 
we  find  a  commensurable  root  ax ,  it  is  advisable  to  depress 
the  equation  by  dividing  by  (x—a^),  since  we  thus  greatly 
facilitate  the  finding  of  the  remaining  roots.  As  these  com- 
mensurable roots  will  present  themselves  in  the  course  of  the 
preliminary  analysis  recommended  in  Chap.  X,  we  consider  it 
unnecessary  to  make  a  special  research  for  such  roots,  which 
are  of  comparatively  rare  occurrence. 


EXERCISES. 

Solve  the  following  equations  : 

1.    x*  -  5  =  0.                         5. 

xv>  _  i  -  o. 

2.    x*  +  2  =  0.                          6. 

»*>  +  3  =  0. 

3.    x5  —  7  =  0.                         7. 

aris  +  l  =  o. 

4.    a*  +  4  =  0.                         8. 

x15  —  20  =  0. 

Depress  the  following  equations  : 

1.    x1  —  1  =  0.                        3. 

211  +  1   =   0. 

2.    a?  +  1  =  0.                         4. 

aj»  — 1  =  0. 

DEPRESSION    OF    EQUATIONS.  75 


CHAPTEK    VII. 

SOLUTION    OF    EQUATIONS    BY    GENERAL    FORMULAS. 

117.  The  only  equations  for  which  general  solutions  have 
been  found  are  those  of  the  first  four  degrees,  and  the  equa- 
tions that  can  be  depressed  to  any  of  those  degrees. 

Leaving  aside  equations  of  the  first  and  second  degrees, 
which  are  fully  treated  of  in  elementary  algebra,  we  propose 
in  the  present  chapter  to  give  the  algebraical  solution  of  cubic 
and  biquadratic  equations. 

CUBIC    EQUATIONS. 

118.  By  the  method  given  in  Art.  71,  any  proposed  cubic 
may  be  reduced  to  the  form, 

x3  +  qx  +  r  —  0.  [1]. 

Assume  that  x  is  the  sum  of  two  other  unknown  quantities, 
that  is,    x  =  y  -\-  z  ; 
then         x3  =  Syz  (y  +  z)  -f-  y3  +  z3 ; 
in  this  replacing  y  +  z  by  x,  and  transposing,  we  have 

2?  _  zyzx  _  (tf  +  z*)  =  0.  [2]. 

In  order  that  [2]  may  be  identical  with  [1]  we  must  have, 

Syz  =  -  q,  (1),  and  f  +  #  =  -  r,  (2). 

Q3 

From  (1)  we  have  y3z3  =  —  ^- ,  and  from  (2),  y3  +  z3 
=  —  r ;  thus  the  sum,  y3  -f  z3,  of  two  unknown  quantities 

being  given,  and  their  product  yh3,  we  can  determine  the 

a3 
quantities  by  the  quadratic  equation  t2  +  rt  —  ^  =  0,  which 

/it 

is  called  the  reducing  equation.     Solving  this  equation,  we 
have 


76  ALGEBRAICAL    EQUATIONS. 


h<>i^=  -s  +  yT  +  n, 


/  r 


r  I  r2        qc 


r  fr2       a3 

-o  -  Y  4 - 


2  " "V  4  "*■  27 ' 


and,  since  a;  =  2/  +  2,  we  have  the  general  formula  for  the 
roots, 

» + w - ^vpi +ti-^i = ■ 

By  Art.  113  the  cube  root  of  y3  may  be  any  of  the  three 
expressions, 

and  the  cube  root  of  z3  any  of  the  three, 

z,    £(_H-</=3)s,    ^(-l-V^S)^ 

where  2/  and  2  are  the  arithmetical  cube  roots  of  y3  and  z3 ; 
there  would,  therefore,  be  nine  values  of  x,  if  we  could  take 
any  one  of  the  three  values  of  tyy3  with  any  one  of  the  three 
values  of  %/z3.  But  one  of  the  conditions  employed  in  ob- 
taining the  reducing  equation  was   yz  =  —J~,  that  is,  the 

product  of  any  pair  of  cube  roots  admissible  in  the  solution 
must  be  a  real  quantity.  It  will  be  seen  that  the  only  pairs  of 
the  cube  roots  that  satisfy  this  condition  are, 

x  =  y  +  z, 

x  =  i(-l  +  V^3)y  +  i(-l-V^3)z  =  -i\(y  +  z)  +  (y-z)V^\, 

x  =  K-i_v^)y+K-i+V=8)i  =  -i\  (y+«)-(jr-*)V^8l. 

and  these  are  the  three  roots  of  x3  -f-  ^  +  r  =  0. 

119.  The  occurrence  of  nine  different  values  of  tyy3  +  %/z3 
is  explained  by  the  fact  that  in  the  process  of  solution  we 

cubed  the  expression  yz  —  -~  .    Now  yh3  maybe  the  cube 

o 

of  any  of  the  expressions  yz,  \  (—  1  +  V—  S)yz,  or 
i (—  1  —  V— 3)  ?/2 ,    so  that   we  really  obtain  the  roots  of 


GENERAL    FORMULAS.  77 

three  different  cubic  equations  in  the  formula.  The  entrance 
of  extraneous  values  into  the  solution  of  an  equation  that  has 
been  squared  to  free  it  from  radicals  is  a  familiar  experience 
in  the  solution  of  quadratic  and  similar  equations. 

120.  We  have  now  to  consider  the  arithmetic  possibility  of 
tho  solution  obtained. 


X  +  $,  >  0,   then  a/ j  +  ^ 


values  of  y  and  z  can  be  determined,  and  the  equation  has 
one  real  root,  y  +  z,  while  the  other  roots  J  {  (y  +  z) 
±  (y  —  z)  V —  3  \  are  evidently  imaginary. 

Ex.  1.  xz  —  6x  —  40  =  0. 


x  =  y  20  +  a/400  —  8  +  j/20  —  V400  —  8 
=   a/39.799..  +    a/. 2.... 

=  3.4142....  +    -5848  =  3.999.... 

We  infer  that  x  =  4  is  a  root,  and  find  upon  trial  that  it 
is  so.  In  this  example  the  remaining  roots  are  most  readily 
found   by   depressing   the    proposed   cubic   to   the   quadratic 

x2  +  4:X  4- 10  =  0,  the  roots  of  which  are  x  =  —  2  ±  a/—  6. 

Ex.  2.  xs  +  12z  +  4  =  0. 


—  y  — 2  -f  a/4  +  64  +   |/ — 2  —  a/4  + 


64 


=   a/6.24621...  +    V— 10  .24621... 

=  1.84165..  -2.17197...  =—.33032... 

The  other  roots  are  most  conveniently  found  by  the  formula; 
they  are  x=  — 1(.  33032...  ±  4.01363.  .V^S). 

(2).   When  —  -f  —■  =  0,  the  formula  reduces  to 

v  /  4       27 


v/^  +  fa  -  *v/^ 


78  ALGEBRAICAL  EQUATIONS. 

In  this  case,  since  y  and  z  are  equal,  the  formula  for  the 
remaining  roots  becomes  x  =  —  \  (y  +  z),  that  is,  there  are 
two  equal  roots. 

Ex.  3.  a?  —  21x  —  54  =  0. 


y  27 


+  V729  -  729  +  Y  27  -  V729  -  729, 
=  2^27  =  6; 
the  remaining  roots  are  x  =  —  3,    x  =  —  3. 

(3).  When  j  +  ^  <  0,  then  W  —  +  Jr   is  impossible, 

and  all  the  expressions  for  the  roots  are  impossible.  But  we 
know  that  the  equation,  being  of  odd  degree,  must  have  at 
least  one  real  root.  Moreover,  the  expression  V#3  must  have 
a  root  of  the  form  m  +  nV—1,  and,  as  z3  differs  from  y3 
only  in  the  sign  of  the  quadratic  radical,  Y&  must  have  a 
root  m  —  nV— '1.      Substituting  these  in  the  expressions 

y  +  %>   —  i  \  (y+z)  ±  (y—z)  ^-^  \>  we  obtain 

277i,    and    —?n±n  Y$  ; 
the  roots  are  therefore  all  real. 

Ex.  4.  x*  —  57s  —  56  =  0. 


x  =  y  28  +  V-6075  +  y  28  —  V-6075 

=  |/28  +  45  V17^  +  j/ 28  —  45  V^ 

We  possess  no  arithmetical  method  of  extracting  these  cube 
roots,  though  it  will  be  found  upon  trial  that 


|/28  +  45a/^3  =  4+a/^3;  and  \/ 28-45V-3  =  4— V-3. 

Thus,  in  the  case  when  the  roots  are  all  real  'and  unequal, 
the  formula  is  practically  unavailable  for  purposes  of  arith- 
metical computation,  since  we  are  not  able  to  perform  the 
operations  indicated.  On  this  account  this  is  sometimes 
called  the  Irreducible  Case  of  Cardan's  Formula. 


GENERAL    FORMULAS.  79 

121.  That  y  and  z  are  necessarily  imaginary  expressions, 
when  all  the  roots  are  real  and  unequal,  and  what  functions 
these  expressions  are  of  the  roots,  may  be  shown  as  follows : 

Since  the  equation  x3  +  qx  +  r  =  0  has  any  one  of  its 
roots  equal  to  the  sum  of  the  other  two  with  contrary  sign, 
the  roots  must  be  of  the  form  2a,  —  (a  +  j3),  —  (a— (5), 
where  a  is  always  real,  while  (3,  the  semidifference  of  two 
of  the  roots,  may  be  real,  zero,  or  imaginary.  Therefore  we 
may  put 

x5  +  qx  +  r  =  x*  —  (3a2  +  j32)z  —  (2a3— 2aj32)  =  0  ; 

from  this  identity  we  have 

3«8  +  0s  =  —  q  (1),       2a3  —  2a/32  —  —r  (2), 

from  which  to  find  a  and  (3  in  terms  of  q  and  r. 

Thus   J  +  jj  =   (a»-o0»)»  -  (a2  +  ^)3 

=   Safip  +  fyfip—Tfap  =  -3(^)3—^3)2. 

£  +  |J  =  ±  v^w-i/33). 


V 


7* 

Adding  —  -  and  its  equivalent  (a3— a(32)  to  the  members 
of  the  above, 


r   ,       /r2 


a       ^3 

"*"  27 


= 

(a?  —  a($)  ±  V- 

-  3  (a2/3 

-W 

= 

as±  V^-a2^- 

-^2=F 

V^f 

-i-*M 


9  ^  r     a 


qZ 


+  h  =  a± 


2"1-  r    4    '27  V-3 

that  is,   y  =  a  -I — —  ,     2  =  a —  . 

V-3  V^3 

Thus  we  find  that  the  expressions  in  Cardan's  Formula  are 

perfect  cubes  in  regard  to   o±  — =  .      The  difficulty  is 

V—  3 


80  ALGEBRAICAL    EQUATIONS. 

that  we  possess  no  arithmetical  method  of  obtaining  these 
cube  roots  in  a  finite  form  even  when  a  and  (3  are  commen- 
surable.    Had  we  any  means  of  obtaining  a  -\ — and 

3  V  — 3 

a ===  ,  their  sum,  2a,  would  give  one  root,  and  their 

V—  3  

difference  multiplied  by  £v  —  3,  would  give  ft  enabling  us 

to  find  the  other  roots  a  ±  ft     We  also  see  : 

(1).  When  (3  is  real,  the  roots  2a,  —  (a  ±  (3),  are  all  real; 

but  — ==   is  imaginary,  and  the  formula  necessarily  assumes 

V  —  3 
the  so-called  irreducible  form. 

(2).  When  (3  is  zero,  the  part  under  the  quadratic  radical 
vanishes,  the  formula  becomes  x  =  V^3  -f-  Va?',  and  the 
roots  are,  2a,  —a,  —a,  i.  e.,  there  are  two  equal  roots. 

(3).  When  (3  is  imaginary,  is  real,  and  there  is  one 

V  —  3 

real  root,  2a,  the  others,   —  (a  ±  (3),  being  imaginary.      In 

this  case,  the  quadratic  radical  in  the  formula  being  possible, 

we  can  find  its  square  root,  and  then   obtain   approximate 

values  for  the  cube  roots  whose  sum  =  x. 

122.  We  have  seen  that  the  criterion  for  the  roots  being 
all  real  is,  that  -r  +  ~ ,  or  27r2  +  4=q%  must  not  be  a  nega- 
tive quantity.  With  the  notation  employed  in  (121),  we  have 
27y2  +  ±(f  -  _  4  (81a4ft  -  18a^  +  ft5")  =  —  4/32(9a2  —  ft*) 
=  — 4ft(3a-f  (3)2(3a— (3)2,  which  last  expression  is  obviously 
the  product  of  the  squares  of  the  differences  of  the  roots. 
Hence,  if  the  roots  2a,  —  (a  ±  (3)  are  all  commensurable, 
the  expression,  27r2  +  4=rf,  must  be  a  perfect  square,  which  is 
the  product  of  at  least  two  squares  different  from  unity.  If 
there  is  but  one  commensurable  root,  it  may  be  shown  that 
the  same  expression  must  have  at  least  one  factor  a  perfect 
square  different  from  unity. 

123.  Many  vain  attempts  have  been  made  to  discover  a 
process  for  obtaining  in  a  finite  form  the  cube  roots  of  ex- 


GENERAL    FORMULAS.  81 

pressioDS  of  the  form  m  +  nV—  h  But  just  as  the  extrac- 
tiou  of  the  square  root  of  such  an  expression  involves  the  solu- 
tion of  a  quadratic  equation,  so  the  extraction  of  the  cube  root 
of  m  +  nV — 1  requires  the  solution  of  a  cubic  equation, 
which  will  be  found  to  be  the  original  cubic  that  led  to  that 
expression. 

124.  The  expression  (m  -f  n  V—  1)*  mav>  &  *s  true,  be 
expanded  in  a  series  by  means  of  the  Binomial  Formula. 
We  thus  obtain  an  approximate  value  of  (m  +  n  V—  1) 3  of 
the  form  P-\-  QV—1 ;  an  approximate  value  of  (m—nV—1)^ 
must  therefore  be  P  —  QV—1;  thus  we  obtain  2P  as  an 
approximate  value  of  x.  This  method,  however,  is  of  no 
practical  use,  as  the  calculations  are  laborious  and  the  series 
are  usually  of  slow  convergence. 

125.  The  roots  of  a  cubic  may  also  be  obtained  by  means 
of  a  table  of  sines  and  cosines ;  we  shall  merely  indicate  the 
process. 

By  trigonometry  we  know  that  the  equation 

cos30  —  |cos  6  —  Jcos  30  =  0,  [1]. 

has  for  roots,  cos  0,  cos  (120°  +  0),  cos  (120°  —  6). 

Now  if  the  proposed  equation  be  xz  —  qx  —  r  =  0,  we 
can,  by  Art.  59,  transform  it  into  another  the  roots  of  which 

are  those  of  the  proposed  each  multijilied  by  A  /  —  ;  thus, 

*-»— Vl?=-a  [3]- 

If  we  assume  [2]  as  identical  with  [1],  we  have 

y  =  cos  (9,  and  ^\/^r^  =  icos30,  or  r\  /  j-g  =  cos  SO. 

As  the  cosine  of  no  angle  can  be  greater  than  unity,  this 
last  equation  can  hold  true  only  when  %V  is  not  greater  than 
4^3,  i.  e.,  when  all  the  roots  are  real.     The  roots  of  [2]  are 


82  ALGEBRAICAL  EQUATIONS. 

therefore  cos0,  cos  (120°+ 0),  cos  (120°— 0) ;  and  those  of  the 
proposed  equation, 

v/y  '  cos  0,     A/y  *  cos  (120°  ±  6). 

To  find  these  roots  we  determine  from  a  table  of  cosines  the 
angle  whose  cosine  =  ?*\/t-§  or  \/  ~TT '  ^is  ang^e  wiU 
be  30,  one-third  of  which  is  0  :   the  cosine  of  0  multiplied  by 

~  will  be  the  greatest  root ;   and  similarly  for  the  other 
o 


V 


roots. 

EXERCISES. 

Solve,  by  Cardan's  Formula,  the  following  equations  : 

1.  s3  +  6x  +  2  =  0. 

2.  a3  —  12s  —  20  =  0. 

3.  s3  +  7s  —  7  =  0. 

4.  s3  +  lis  +  8  =  0. 

5.  5s3  —  13s  —  52  =  0. 

6.  lis3  —  56s  +  105  =  0. 

126.  In  conclusion  it  may  be  observed  that  all  attempts  at 
an  algebraical  solution  of  the  cubic,  lead  to  a  formula  similar  to. 
or  identical  with,  that  of  Cardan.  Among  others,  Tschirnhau- 
sen,  Tischendorf  and  Lagrange,  by  very  different  paths,  arrive 
at  the  same  result.  We  shall  indicate  the  following  investiga- 
tion, which  the  student  may  work  out  as  an  exercise.  It  may 
be  easily  shown  that  the  equation  s3  +  3s  —  (az—a~s)  =  0  has 

a  root  a .    Any  proposed  cubic  s3  -f-  qx  +  r  =  0    may 

a  /27 

(59)  be  transformed  to   yz  +  ?>y  +  r\  /  —^  =  0.     We   have 

now  to  find  a  from  the  equation  as  —  a~3  =  —  r  \  /  —  ,  or 
a?  +  A  /  — g-  •  az  —  1  =  0,  and  this,  we  find,  leads  to  a  for- 
mula for  s  similar  to  that  of  Cardan. 


GENERAL    FORMULAS.  83 

In  a  subsequent  chapter  we  shall  take  up  the  solution  of 
eubics  by  Horner's  Method,  and  it  will  be  seen  that  the  com- 
putation of  a  root  of  a  cubic  of  the  form  x3  +  qx  +  r  —  0 
is  a  matter  of  no  more  difficulty  than  the  arithmetical  extrac- 
tion of  a  cube  root. 


BIQUADRATIC    EQUATIONS. 

127.  Of  the  various  algebraical  solutions  of  the  biquadratic 
that  have  been  proposed,  the  first,  historically,  is  that  discov- 
ered by  Ferrari,  a  pupil  of  Cardan's,  soon  after  the  publication 
of  that  solution  of  the  cubic  known  as  Cardan's. 

Ferrari's  Solution.  —  Let  the  proposed  equation,  de- 
prived of  its  second  term,  be   a4  -f  qx2  +  rx  +s  —  0,   then 

x*  =   —  qx2  —  rx  — 5. 

Add  to  both  sides  of  the  equation  2Jcx2  +  I2,  then 

(x2  +  hf  =  {2k  -  q)  x2  -  rx  +  (*?  -  *)■ 

We  have  now  to  determine  h  so  that  the  second  member 
may  be  a  complete  square.     In  order  that  this  may  be  the  case, 

we  must  have  (21c— q)  (h2—s)  =  ~r  >  or 

£3  _  Jgjp  _  fa  _|_  (Lqs  _  lr2)   -  0. 

From  this,  the  so-called  reducing  cubic,  having  determined 
a  value  of  k,  by  any  of  the  methods  given  for  the  solution  of 
eubics,  the  solution  of  the  proposed  biquadratic  is  reduced  to 
that  of  the  two  quadratics, 


x2  -f  h  =       V2h—q  '  x  —  Vh2  —  s, 
x2  -f  h  =  -  \/2k—q  '  x  -f-  \/lc2  —  s. 

The  following  solution  is,  however,   generally  more  con- 
venient. 

128.  Descartes'  Solution.— As  before,  let  the  proposed 
equation  be 

34    _|_    gX2    +    rZ    +    S    —    0,  [1] 


84  ALGEBRAICAL  EQUATIONS. 

and  assume  the  first  member  to  be  the  product  of  two  qua- 
dratic factors  (x2  -f  hx  +  /)  and  (x2  —  lex  -f  g),  thus, 

*  +  if  +  g-V)^  +  {gh-fk)x+fg  =  0.      [2]. 

Equating  coefficients  of  equal  powers  of  x  in  [1]  and  [2], 
we  obtain  the  equations, 

f  +  ff—&=  q>    gk—fk—  r,   fg  =  s.  [3]. 

Having  found  /  and  g  in  terms  of  h  from  the  first  two  of 
these  equations,  and  substituting  in  the  third,  we  have 

(W+l  +  q)(W-l  +  q)=is, 
from  which  we  obtain  by  reduction, 

£6  +  2q&  +  (q2  -  4s)  h2  -  r2  =  0, 
or,  putting  z  for  h2,  we  have  the  reducing  cubic, 

zz  +  2qz2  +  (q2  —  4:s)z  —  r2  =  0. 

"When  from  this  cubic  a  value  of  z  has  been  determined,  its 
square  root  is  h,  and  k  being  known,  /  and  g  become  known 
from  the  first  two  of  the  equations  in  [3],  and  we  obtain 
the  four  roots  of  the  biquadratic  from  the  two  quadratics, 
x2  +  hx  +  /  =  0,    x2  —  kx  +  g  =  0. 

Ex.  x*  —  2±x2  +  15x  —  2  =  0. 

The  reducing  cubic  is, 

#  _  48^2  +  5842  _  225  =  0. 

From  this  we  find  one  root,  z  =  25,  .\  h  =  5,  /  =  2, 
g  =  —  1 ;  and  we  can  determine  the  four  roots  of  the  pro- 
posed from  the  equations,  x2  +  bx  +  2  =  0,   x2  —  hx  —  1  =  0. 

129.  We  shall  give  one  more  solution,  of  some  interest  as 
proceeding  by  a  method  analogous  to  that  pursued  in  Cardan's 
solution  of  the  cubic ;  a  method  that  has  been  applied  to  the 
solution  of  equations  of  higher  degree  than  the  fourth,  and 
fails  only  in  so  far  as  that  the  reducing  equations  obtained  are 
of  higher  degree  than  the  original  equation. 


GENERAL    FORMULAS.  85 

130.   Euler's  Solution". — Let  the  proposed  equation  be, 

#4  _j_  qX2  +  rx  _|_  s  —  0#  j-^ 

Assume  #  =  y  +  z  -\-u,    then 

3,2  —  ^2  _|_  z2  _j_  ^2  _|_  2  (^  -f  yw  4-  zu), 
and  x2  —  y2  —  z2  —  u2  =  2(yz  +  yu  +  2^). 

Squaring  both  sides,  we  obtain 

•^  —  2(y2  +  z2  +  u2)x2  +  (#2  +  22  +  w2)2  =  4:(yz  +  yu  +  zuf 

=*Ky2z2  +  2/%2  +  z2u2)  -f  8^?«  (?/  +  2;+  u) 

Replacing  y  +  z  +  u  by  a;,  and  transposing,  we  have 

x*  —  2(y2  +  z2  +  u2)  x2  —  8yzu  ■  a;  +  (?/2  +  z2  +  w2) 

—  4  (?/2;z2  +  #%2  +  z2u2)  =  0.      [2]. 

In  order  that  [2]  may  be  identical  with  [1],  we  must  have 
q  =  —  2  (y2  +  z2  +  ^2) ;        r  ==  —  Syzu  ; 
5  =  (3/2  +  z2  +  w2)"—  4  (yh2  +  ?/2^2  +  A2), 

or,  tf  +  z2+u2=-lr;    y2z2  +  tj2u2  +  z2it2  =  l(f-s)  =  ^^; 
/£  4  \4        /  lo 

r2 

W22%2    =    — -. 

*  64 

The  conditions  show  (45)  that  y2,  z2,  u2,  must  be  the  roots 
of  the  cubic  equation 

1  +  r  +    i6      *~64  -  °* 

Denoting  the  roots  of  this  equation  by  tl9  t2,  h  >  then 

#  =  ±  VS ,       «  =  ±  V^2        w  =  ±  V5 . 

If  we  take    x=y  +  z  +  u=±   V~ii  ±  V~t2  ±  V^3, 
we  obtain  eight  different  values  for  x.     But  some  of  these  are 

inadmissible,  since  one  of  the  conditions  was    yzu  =  —  —  , 

8 
so  that,  supposing  r  to  be  positive,  the  only  admissible  values 
of  x  are, 

— Vh—Vk—Vh,  —Vti+Vh+Vk,  Vh—Vh+Vh, 
Vk+Vh-Vh, 


86  ALGEBRAICAL    EQUATIONS. 

namely  those  combinations  the  product  of  whose   terms  is 
negative. 
If  r  is  negative,  the  admissible  values  of  a  are, 

Vh+Vl+Vh,   Vti-Vt-z-Vt-s,  -Vh+Viz-Vtz, 
—Vti—Vtt+Vh. 

131.  The  occurrence  of  the  eight  values  of  a  in  the  solu- 
tion has  an  explanation  similar  to  that  given  for  the  appear- 
ance of  nine  values  for  x  in  the  solution  by  Cardan's  For- 
mula  (119).      In  the   course  of  the  solution  we  employed 

^»2  ^2 

ifzhi2  =  —  ;    now  —   may  arise  from  r  either  positive  or 

negative,   so   that   the    solution    contains   the    roots   of   the 
equation, 

&  _l_  qXi  —  rx  +  s  =  0, 

us  well  as  those  of  the  proposed  equation. 

It  may  be  remarked  that  the  reducing  cubic  found  by 
Euler's  Method  will  be  found  to  coincide  with  that  obtained 
by  Descartes'  Method,  if  we  put  U  =  h2. 

EXERCISES. 

1.  a4  —  22a2  —  21a  +  30  =  0. 

2.  a4  —  7a2  +  26x  —  40  =  0. 

3.  a4  +  3a2  +  2x  +  3  =  0. 

4.  a4  —  12a3  +  40a2  —  29a  +  6  =  0. 

5.  5a4  +  22a3  +  17a2  -  36a  -  21  =  0. 

132.  The  preceding  methods  of  solution  depend  upon 
special  analytical  artifices,  and  have  no  apparent  bond  of 
union  with  each  other,  or  with  the  methods  of  solution  em- 
ployed for  equations  of  lower  degrees.  By  the  method  of 
investigation  employed  below  it  will  appear  that  the  algebraical 
solution  of  the  quadratic,  and  all  those  given  for  the  biquadratic 
above,  are  but  particular  cases  of  the  solution  of  the  general 
problem:  to  reduce  the  solution  of  an  equation  of  the  2nth  de- 
gree, to  that  of  two  equations  of  the  nth  degree. 

Let  us  suppose  /(a)  =  0,  which  is  of  the  2;Jh  degree,  to  be 


GENERAL    FORMULAS.  87 

the  product  of  two  factors,  <p  (x)  and  i/>  (x),  each  of  the  ntk  de- 
gree, then 

Every  function  of  even  degree  may,  therefore,  be  regarded 
as  being  the  difference  of  two  functions,  which  are  complete 
squares.  If  we  can,  then,  resolve  f(x)  into  the  difference  of 
two  squares,  we  can  reduce  the  solution  of  f(x)  =0  to  that 
of  two  equations  of  one-half  the  degree  of  f(x).  For,  trans- 
posing one  of  the  squares  in  [1],  and  extracting  the  square 
root  of  both  sides  of  the  resulting  equation,  we  have 

<P  (s)  +  V>  (x)  _    ,  <t>{x)  —  i>  (x) 

2  ~   *  2 

from  which  we  obtain,  according  as  we  take  the  upper  or 
lower  sign  with  the  second  member, 

<p(x)  =  0,    or    *p(x)  =  0, 

two  equations  of  one-half  the  degree  of  f(x)  =  0. 

133.  Theoretically  this  may  be  done ;  but  we  can  see  in 
advance  that,  for  equations  of  above  the  fourth  degree,  the 
reducing  equations  will  be  of  higher  degree  than  the  proposed 
equation.     For,  from  the  %n  binomial  factors  of  f(x)  may  be 

\2n 
formed  7=-  factors  0  (x)  and  \1>  (x),  each  of  the  nth  degree 
\n\n  T  v  '  r  \  j> 

(45),  hence  from  f(x)  we  may  form  the  expressions  <f>(x)  ±\j)(x) 

\2n 
in  if^f-  different  ways.      It  is  evident,  therefore,  that  the 

coefficients  of  the  functions    (p(x)  ±  ^p(x)  may  have  $-. — =- 

different  values,  and  will,  consequently,  have  to  be  determined 
by  equations  of  that  degree.  To  find  the  degree  of  the  re- 
ducing equations  for  equations  of  the  second,  fourth,  and 
sixth  degrees  respectively,  we  substitute  successively  1,  2,  and 

\%n 
3,  for  n  in  nl    7  ,  and  obtain  1,  3,  and  10,  as  the  numbers 

2\n\n 

expressiug  the  degrees  of  the  respective  reducing  equations. 


88  ALGEBRAICAL  EQUATIONS. 

The  quadratic  and  biquadratic  may  therefore  be  reduced  by 
means  of  reducing  equations  of  lower  degree ;  but  to  solve  a 
bicubic  equation  by  means  of  two  cubics,  we  should,  except 
in  particular  cases  (145),  have  first  to  form  and  solve  a  re- 
ducing equation  of  the  tenth  degree. 

134.  To  illustrate  the  process  of  obtaining  the  reducing 
equation,  it  will  be  sufficient  to  suppose  f(x)  of  the  sixth  de- 
gree ;  then  </>  (x)  and  i/>  (x)  will  be  each  of  the  third  degree. 

Let       <p  (x)  =  x3  +  mYx2  +  m2x  -f  w3, 

ip(x)  =  x3  +  nx  x2  +  n2  x  +  n3 , 


"     I         2  j  -  x  +       2      X  + 

(0(g)— #g)j  =  \mx-nX0  [   m2-%^  [   w3-%) 

Changing  17"  ,  &c,  to  A,  B,  C,  and  — ^r — -,  &c,  to 
«,  b,  c,  and  equating,  we  have,  since  (131)  <f>  (x)  -f  ip  (x) 
=  ±\4>(z) -*!>(*)}, 

x?  +  Ax2  +  Bx—  C  =   ±  (ax2  +  bx  +  c). 

Squaring  both  sides,  and  collecting  terms,  we  have 

x*  +  2  Ax5  +  (A2  —  ^  +  2B)  x*  +  (2^15  —  2«6  +  2(7)  g3 
+  (^2  _  p  +  2  J  C-  2ac)  x2  +  (250  —  21c)  x 
+  (C*  -  c2)  =  0,  [1], 

the  equation  of  the  sixtb  degree  having  its  coefficients  ex- 
pressed as  functions  of  the  coefficients  of  7T  ■    From 

this  we  may  deduce  the  corresponding  equations  of  the  fourth 
and  second  degree  by  omitting  the  letters  that  become  zero; 
thus  we  obtain, 

«4  +  2Ax*  +  (A2  -a2  +  2B)  x2  +  (2AB  -  2ah)  x 

+  {B2-V)  =  0,     [2] 
x2  +  2Ax  +  (A2  —  a2)  =  0,  [3]. 


GENERAL    FORMULAS.  89 

135.  We  obtain  the  reducing  equations  by  equating  the 
coefficients  of  [1].  [2],  or  [3]  with  p,  q,  r,  8,  &c,  the  co- 
efficients of  like  powers  of  x  in  the  general  equations.  Ex- 
cept A  (which  always  ==  \p),  the  coefficients  B,  C,  &c,  of 

0  will  have  to  be  determined,  as  shown  above,  by 

|2» 
equations  of  the  kth  degree,  where  h  =  -pf- ,  while   a,  b,  c, 

&  (%\  J)  (%\  L_  I— 

the  coefficients  of   ±  0  ,  will,  since  they  have  the 

double  sign,  be  involved  in  equations  of  the  form  (y2)k  +  P(y2)k~l 
+  Q  {y2)k~2  +  &c.  =  0,  which  may  also  be  solved  as  equations 
of  the  l4h  degree. 

136.  It  will  be  observed  that,  since  in  the  equations  [1], 

[2],  [3],  there  are  2n  —  1  unknown  quantities,  B,   C, 

a,  b,  c,  . . . .  there  will  be  2)i  —  1  forms  of  the  reducing  equa- 
tion, according  as  we  eliminate  for  one  or  other  of  the  un- 
knowns ;  it  will  also  be  seen  that  one  of  these  equations  being 
formed,  the  others  may  be  deduced  from  it  by  suitable  substi- 
tutions. Thus  there  is  but  one  form  of  the  reducing  equation 
for  the  quadratic,  there  are  three  distinct  forms  for  the  bi- 
quadratic, two  of  which  have  been  given,  and  five  for  the 
bicubic. 

137.  We  shall  first  find,  by  this  method,  the  reducing 
equation  for  the  quadratic. 

Comparing  the  two  forms  of  the  quadratic, 

x2  -f-  px  4-  q  =  0, 

x2  4.  2Ax  4-  (A2  — a2)  =  0, 

and  equating  corresponding  coefficients,  we  have 

A  =  | ;    4s  -  «'  =  % .-  a?  =  q ', 

))2  4:(7 

whence  a2  =  ±  - — j — - ,  which  is  the  reducing  equation,  and 

,         „      <p(x)  —  V  (z)  ,    0  (x)  —  xb  (x) 

we  have  for  rv  ;       rv  '  =   ±     v  7       yv  ; 


p  _  Vp2-±q 

X+  2  -   ±         2 


90  ALGEBRAICAL  EQUATIONS. 

the  usual  form  to  which  we  reduce  the  quadratic,  and  equiva- 
lent to  two  equations  of  the  first  degree. 

138.  Again,  comparing  the  equation  of  the  fourth  degree, 

#4  -f-  pxz  -f-  qx2  -f-  rx  -f  s  =  0, 
and    x*+2Ax*+(A2-a*  +  2B)x*+(2AB-2a1))x+(BZ-&)=  0, 

we  have,  A=%;  A2—a*-h2B  =  q;  2AB-2ab  =  r;  B*-b2  =  s. 
From  the  second  of  these  conditions  we  have 
«a  =    ^  +  2B-q, 

-     - = ew  ■  (4fc=)" 

from  the  third  and  fourth  ;   thence 

f       oz?  p2B2-2prB  +  ri      , 

4L  +  2B  -  q  =         4(jB/lg) ,  whence 

8^3  _  4^  +  (2pr  _  8s)5  —  (^  _  4^s  +  r2)  _  0     [!]  . 

from  which,  if  we  put  p  =  0,  we  obtain 

SBS  —  ±qB*  —  8sB  +  (iqs  —  r2)  =  0, 

the  reducing  cubic  as  found  by  Ferrari. 

139.  The  reducing  cubic  involving  a  may  be  found  by 
elimination,  as  was  the  preceding  one  ;  but  it  is  more  readily 
found  by  substituting  in  [1]  the  value  of  B  in  terms  of  a. 
The  full  equation  thus  obtained  is, 

a«  +  [%q  -  ^)  a"  +  (^4  -  fq  -f  pr  +  qt-  4s)  a2 

_  /i3  -  ^  +  8?y  =  0 

which  becomes,  putting  p  —  0, 

«6  +  2</«4  +  (g»  —  4s)  a2  —  r2  =  0,  [2] 

the  reducing  cubic  as  found  by  Descartes  and  Euler. 


GENERAL    FORMULAS.  91 

140.  The  reducing  cubic  involving  b  may  either  be  found 
directly,  or  by  substitution  from  the  others ;   it  is, 

64£6  +  (32pr  +  64s— Wq2)^  +  (Apir2—32prs—8psqs—8q?>2)tf 

_  (r*-2p2r2s-p**2)  =  o, 

which  putting  p  =  0,  and  40s  =  z,  may  be  simplified  to 
23  +  (4g  _  g2)  ^  _  2^  _  r4   _    o. 

Having  obtained  any  one  root  of  any  of  the  above  reducing 
eubics,  we  can  find  the  quantities  A,  B,  a,  b,  from  the  relations 

2  A  =  p,  2B  =  a2  -f  q  —  A2;  2b  =  P  ~~"r>  and  then  put 
the  equation  in  the  form, 

<f>(x)  +  V>(aQ   _     ,    <j>{x)-^{x) 

2         "  "    ±  2 

that  is,  x2  +  Ax  +  B  =   ±  (ax  -f  0), 

and  solve  by  two  quadratic  equations. 

141.  We  may  examine  what  functions  the  roots  of  the  re- 
ducing eubics  are  of  those  of  the  biquadratic. 

Supposing  the  roots  of  the  biquadratic  to  be  a,  (3,  y,  d, 
and  those  of  the  reducing  cubic  of  Ferrari,  BY,  B2,  Bz, 
we  have 

_  m,  +  n2  _  aP  +  yd  ay  +  fid  ad  +  fiy 

*i-— g—  -— tj— ,    ^2  =  —^—,    Z?3  =  — g— ; 

that  is,  the  roots  of  Ferrari's  reducing  cubic  are  the  three 
possible  functions  of  the  form  T"  —  .  Similarly  the  roots 
of  the  reducing  equation  in  b  are,  since  b  —     2~    2, 

_  a(3  —  yd             _  ay  —  fid                 ad  —  (3y 
0i  —  s ,     02  —  n •     #3  = « • 


Denoting  the  roots  of  Descartes'  reducing  cubic  by  ax2,  a22, 
a32,  then,  since  (133)  a  =  — ~ — - ,  where  —  m  denotes  the 


02  ALGEBRAICAL   EQUATIONS. 

sum  of  any  two  of  the  roots,  ami  —  n  the  sum  of  the  remain- 
ing two,  we  have, 

±ax  =  =F  h(a  +  P-y-6),     ±a2  =  =F  i(o-i3  +  y-d), 
±(h=  =Fi(a-/3-y  +  d),  also  .4  =  -  £(a  +  0  +  y  +  d), 
...    K—^i—^—^—^)  —  ttj      j(_^_ai  +  a2  +  fl53)  _  p9 
i(— A+av— a2  +  (h)  =  y,      £(— A  +  a^Oz— «3)  =  <J. 

If  we  suppose  A  =  0,   2*.  e.,  ;;  =  0,  we  obtain 
i  (—  fli  —  «2  —  «3)   =   «>    &c., 
the  same  results  we  obtained  in  (130). 

142.  If  the  biquadratic  has  all  its  roots  real,  it  is  obvious, 
from  consideration  of  what  function  of  these  the  roots  of  the 
reducing  equations  are,  that  the  latter  are  also  all  real. 

If  the  roots  of  the  biquadratic  are  all  imaginary,  they  must 
be  of  the  form  P  ±  QV^l  and-iZ  ±  SV^I,  and  upon 
putting  these  for  a,  (3,  y,  6y  in  the  expressions, 

a(S  +  yd                                  /a[3-yd\2 
B\  =  j-*-*        '        bl    =  \ 2"/'        ' 

2         (a  +  P-y-dV    o 

we  find  that  all  the  roots  of  the  reducing  cubics  are  real  in 
this  case  also. 

If  the  biquadratic  has  two  real,  and  two  imaginary  roots,  we 
find,  in  the  same  way,  that  in  this  case  each  of  the  reducing 
cubics  has  two  imaginary  roots,  or,  possibly,  two  equal  roots ; 
that  is,  the  reducing  cubic  will  not  fall  under  the  irreducible 
case  when  the  biquadratic  has  two  real  and  two  imaginary 
roots. 

143.  Upon  consideration  of  the  final  terms  of  the  different 
reducing  cubics,  we  find  that,  if  we  have  either  p2s  —  ±qs 
+  r2  =  0,  jp  —  fyq  +  8r  =  0,  or  r4  —  2]j2r2s  —  7A2  =  0, 
the  biquadratic  £4  -f  px*  +  qx2  +  rx  -f  s  =  0  may  be  solved 
at  once  by  means  of  quadratic  equations,  for  this  is  equivalent 


GENERAL    FORMULAS.  93 

to  having  either  B{  =  0,  or  bv  =  0,  or  av  =  0,  and  we  pro- 
ceed as  in  (140). 

Ex.  1.        a4  —  W  —  60a2  +  221a  —  1G9  =  0. 

Here  we  find  p2s  —  ±qs  +  r2  =  0.     We  proceed  to  reduce 

7)         7 
the  equation  to  two  quadratics  as  follows :    A  =  -£  =  •= , 

B1  =  0,     *>  =  A*  +  25  -  gr   =  ~  +  0  +  CO  =  -^ J 

17 
.-.  Oi  =  ±  -jr ;    K1  =  Bx2  —  s  =  1G9,   .\  J£  =  ±  13  ;   hence 

we  have, 

a:2_^  +  0    =     ±(^S  +  13), 

therefore  we  have 

a?  _  22a;  +  13  —  o,    and    a2  -f  5a;  —  13  =  0. 

Ex.  2.        a4  —  14a8  +  63a2  —  98a  +  24  =  0. 

Here  we  find  p%  —  kpq  -f  8r  =  0,  therefore  ax  =  0.  Pro- 
ceeding as  before,  we  find  A  =  —  7,  B  =  7,  #i  =  ±  5, 
and  have  the  two  equations, 

^2  _  7^  _+_  7  =   ±  5. 

Ex.  3.  a4  +  3a3  —  4a2  +  9a  +  9  =  0. 

Here  wTe  find  r4  —  2p2r2s  — p4s2  =  0,    therefore   bx  =  0, 

7  3  7 

5  =  3,    «  =  ±  - ;   hence  we  have  a2  +  ^  a  +  3  =  ±  ~  a  ; 

therefore 

a;2  _  9X  +  3  _  o,    and    a2  -f  5a  +  3  =  0. 

144.   A  cubic  equation  may  be  put  in  the  form, 

a4  +  px3  -f  qx2  +  rx  =  0, 

that  is,  may  be  regarded  as  a  biquadratic  having  one  root 
x  =  0.  We  may,  accordingly,  derive  reducing  equations  from 
those  given  above  for  the  biquadratic  by  simply  leaving  out 
the  symbol  s.  We  thus  find  the  reducing  equation  for  a  cubic, 
found  by  this  method,  is  itself  a  cubic.     We  also  find  that 


94  ALGEBRAICAL    EQUATIONS. 

when  iii  a  cubic  equation,  x*  +  px*  -f-  qx  +  r  =  0,  the  rela- 
tion p3  —  kpq  +  8r  =  0  holds  good,  the  equation  may  be 
reduced  as  in  (143). 

145.  When  we  attempt  to  form  the  reducing  equation  for 
the  bicubic,  we  obtain,  as  we  might  expect,  an  equation  of  the 
10th  degree.  Thus,  like  all  others  that  have  been  tried,  this 
method  fails  to  reduce  equations  of  the  fifth  or  higher  degrees 
by  means  of  reducing  equations  of  lower  degree  than  the 
proposed. 

When  special  relations  exist  among  the  roots,  reduction  may 
be  effected  by  equations  much  lower  in  degree.  Thus  in  the 
equation,  as  found  in  (134), 

z*  +  2Ax5  +  (A*  -a  +  2B)x*  +  {2AB  -  2ab  +  C)x* 

+  (B*—tf  +  2AC-2ac)x*+(2BC—  2bc)x+(C*-c*)  =  0, 

if  we  put  a  =  0,  i.  e.,  suppose  that  of  the  roots  the  sum  of 
one-half  the  number  is  equal  to  the  sum  of  the  other  half,  we 
see  by  inspection  of  the  coefficients  above  that  the  quantities 
A,  B,  C,  b,  and  c,  may  then  be  found  by  the  series  of  simple 
equations, 

A=ip;    B  =  i(4tq-p*);     C=i(r-pB);    a  =  0; 

I  =.  VB*  +  pC—s  ;    c  =  VC*  —  v, 

that  is,  if  in  any  equation  of  degree  not  greater  than  the  sixth, 
the  sum  of  one-half  the  number  of  roots  be  equal  to  the  sum 
of  the  other  half  the  equation  may  be  reduced  by  inspection. 

Ex.  Let  x«  —  10z5  +  22^  +  11a;3  —  50x*  —  U2x  —  96  =  0 
be  an  equation  having  the  sum  of  one-half  its  roots  =  5,  then, 
by  the  simple  equations  above  given,  we  find 

A  =  -5,  B=~\,  C=-2,  <z  =  0,  2>  =  ±y,  e  =  ±10, 
and  we  can  put  the  equation  in  the  form, 

a*  _  5x*  -  |  a;  -  2  =   ±  (~x  +  10 V 
whence 

a*  —  5x2  —  10z  —  12  =  0,    and    x3  —  5x2  -f  7x  +  8  =  0. 


GENERAL    FORMULAS.  95 

If,  in  an  equation  of  the  fifth  degree,  the  sum  of  three  of  the 
roots  is  equal  to  the  sum  of  the  remaining  two,  we  can,  in  the 
same  way,  reduce  the  solution  to  that  of  a  cubic  and  a  qua- 
dratic equation. 

Ex.  2.  Let  a5  +  14a*  +  43a3  —  48a3  —  258a  —  72  =  0 
be  an  equation  having  three  of  its  roots  equal  to  the  other  two ; 
then,  as  in  the  preceding  example,  we  find 

A  =  7,  B  =  —  3,  C  =  —  3,  a  =  0,  b  =  ±  15,  c  =  ±  3, 

and  we  can  put  the  equation  under  the  form, 

xs  +  ;^2  _  3z  _  3  =  ±  (15a;  +  3)? 

whence  we  obtain  the  two  equations, 

a*  +  7^2  _  iqx  _  e  =  o,    and    a3  +  7a  +  12  =  0. 

EXERCISES. 

Depress  the  following  equations  by  the  methods  exemplified 
in  (143)  and  (145) : 

1.  a4  +  2a3  —  3a3  —  92a  —  529  =  0. 

2.  a4  —  17a3  +  30a3  —  195a  —  225  =  0. 

3.  a4  —  8a3  +  20a3  —  16a  —  21  =  0. 

4.  a4  —  20a3  +  117a2  —  170a  —  60  =  0. 

5.  a4  +  7a3  —  20a3  +  35a  +  25  =  0. 

6.  a4  -  3a3  H   0a3  +  93a  —  961   =  0. 

7.  a5  +  24a4  +  142a3  —  29a3  —  543a  —  105  =  0. 

8.  a6  —  W  +  9a4  —16a3  —  40a  —  25  =  0. 


96  ALGEBRAICAL  EQUATIONS. 


CHAPTER    VIII. 

sturm's    theorem. 

146.  In  a  preceding  chapter  were  given  several  theorems 
regarding  the  limits  between  which  the  real  roots  of  an  equa- 
tion must  be  found.  Allusion  was  also  made  to  methods  pro- 
posed by  Eolle  and  Waring  for  separating  the  roots.  The  ob- 
jection to  these  methods  was  not  only  the  excessive  labor 
required  in  their  practical  application,  but  also  the  fact  that 
they  leave  the  number  of  real  roots  existing  in  an  equation 
doubtful  till  we  have  actually  ascertained  the  situation  of 
each. 

The  method  about  to  be  explained  is  the  only  one  hitherto 
proposed  by  which  the  number  of  real  roots  can  be  deter- 
mined a  priori.  Comprising  the  solution  of  a  problem  that 
had  engaged  the  attention  of  analysts  for  above  two  centuries, 
this  method  is  distinguished  as  much  by  the  simplicity  of  the 
process  as  by  the  exactitude  of  the  results  attained. 

147.  Sturm's  Functions.  —  Def.  —  Let   X=0    be  an 

equation  of  the  nth  degree,  having  no  equal  roots,*  XY  the 

first  derived  function  of  X,  and  X2,  X3 , Xn ,  the  series 

of  remainders  obtained  in  performing  the  operation  of  obtain- 
ing the  greatest  common  measure  of  X  and  XY ,  care  being 
taken  to  change  the  signs  of  each  remainder  before  employing 
it  as  a  divisor,f  changing  the  sign  of  the  last  remainder  also ; 
the  series  of  functions, 

X,  Xi,  X2,  X5, ....  x„, 

we  shall  refer  to  as  Sturm's  Functions. 


*  If  the  proposed  equation  have  equal  roots,  the  fact  will  be  discovered  (92)  by  our 
arriving  at  a  zero  remainder  in  the  course  of  the  process.  It  will  be  seen  (156)  that, 
in  such  a  case,  the  theorem  holds  good  for  the  preceding  remainders. 

t  In  the  ordinary  operation  for  the  G.  C  M.  it  is  merely  a  matter  of  convenience 


STURM'S    THEOREM.  97 

148.  Sturm's  Theorem.  —  Let  a  and  (3  be  two  numbers, 
such  that  a  <  (3,  then  the  excess  in  the  number  of  variations 
of  sign  in  the  series  of  functions, 

X,  X\ ,  X2 ,  Xz ,  . . . .  Xn , 

when  a  is  substituted  for  x,  over  the  number  of  variations 
when  (3  is  put  for  x,  expresses  the  number  of  real  roots  of  the 
equation  X  =  0  that  lie  between  a  and  (3. 

Let  Qi ,  Q2, Qn-i ,  denote  the  quotients  obtained  in  the 

successive  divisions ;  then,  since  the  dividend  is  equal  to  the 
product  of  divisor  and  quotient,  plus  the  remainder,  or  minus 
the  remainder  with  its  sign  changed,  we  have  the  following 
relations  existing  among  the  functions  : 

X       =     ft     X1      -    X2,  (1). 

Xx    =    Q2  X2    -  X3,  (2). 

X2    =    &  X3    -  Xi}  (3). 

Xr-\    =      Qr     Xr       —     Xr+i,  (r). 

Xn.2=    Q^Xn-i-  Xn,  (n-1). 

From  these  equations  we  deduce  the  following  conclusions : 

I.  Tivo  consecutive  functions  cannot  vanish  simultaneously 
for  any  value  of  x.  For,  if  possible,  suppose  Xx  =  0  and 
X2  =  0  at  the  same  time.  Then,  by  Eq.  (2),  we  shall  have 
Xs  =  0 ;  and  if  X2  =  0  and  X3  =  0,  then,  by  Eq.  (3),  we 
have  also  X4  —  0,  and  so  on  to  the  last  equation,  which  will 
give  Xn  =  0;  but  this  is  impossible,  since,  by  supposition, 
X  =  0  has  no  equal  roots,  and  therefore  Xn  must  be  a  nu- 
merical remainder,  independent  of  x. 

II.  When  any  function,  except  X,  vanishes,  the  adjacent 
functions  have  contrary  signs.  Suppose,  for  example,  Xr  in 
Eq.  (r)  vanishes,  then  we  have  Xr.x  =  —  Xr+l . 

whether  we  do,  or  do  not,  change  a  remainder  before  employing  it  as  a  divisor  ;  here 
this  change  is  an  essential  part  of  the  process.  Subject  to  this  condition  we  may 
multiply  or  divide  a  remainder  by  any  convenient  number  as  in  the  ordinary  process. 

5 


98  ALGEBRAICAL  EQUATIONS. 

III.  X  and  Xx  have  contrary  signs  immediately  before  X 
vanishes  for  some  value  of  x,  as  x  =  c,  and  immediately  after, 
X  and  Xi  have  the  same  sign. 

Denote  X  and  Xx  by  f(x)  and  fx  (x) ;  put  c  ±  h  for  x, 
then  (7) 

/  [c  ±  h)  =f(c)±  hf  (c)  +  &c,  and  /,  (c  ±  h)  =  f(c)  ±  &c, 

and  h  may  be  taken  so  small  that  the  signs  of  f(c  ±  h)  and 
fi  (c  ±  h)  will  be  the  same  as  those  of  the  first  term  in  their 
expansions  that  does  not  vanish.  But  /(c)  =  0  by  suppo- 
sition, while,  by  I,  /  (c)  cannot  vanish  at  the  same  time. 
Therefore,  when  h  is  taken  small  enough,  the  signs  of  f(c±h) 
and  /i  (c  ±  h)  are  the  same  as  those  of  ±  hf  (c),  and  /  (c) 
respectively,  and  these  differ  in  sign  when  h  is  negative,  and 
vice  versa;  that  is,  X  and  X!  have  contrary  signs  when 
x  =  c  —  h,  and  the  same  sign  when  x  =  c  +  h,  h  being  a 
quantity  as  small  as  we  please. 

Hence,  as  x  increases  through  c,  a  root  of  X  =  0,  one 
variation  of  sign  is  lost  in  the  series  of  Sturm's  Functions. 

IV.  No  variation  of  sign  is  either  lost  or  gained  when  any 
function,  except  X,  vanishes. 

For  suppose  Xr  to  vanish  for  some  value  of  x,  then,  by  II, 
Xr_i  and  Xr+]  have  contrary  signs,  and  with  whichever  of  the 
two  Xr  agreed  in  sign  just  before  it  vanished,  it  will  agree 
with  the  other  after  it  has  passed  through  zero,  and  changed 
sign ;  thus  no  variation  of  sign  is  either  lost  or  gained  when 
Xr  passes  through  zero. 

By  I  no  two  consecutive  functions  can  vanish  at  the  same 
time.  If  two  or  more,  that  are  not  consecutive,  vanish  for  the 
same  value  of  x,  then,  if  X  be  one  of  them,  it  follows,  by  III, 
that  a  variation  of  sign  is  lost  as  x  increases  through  that 
value ;  and,  if  X  is  not  one  of  them,  by  IV  no  variation  is 
lost,  or  gained.  Thus  we  see  that  Sturm's  Functions  never 
lose  a  variation  of  sign  except  when  x  increases  through  a 
root  of  X  =  0,  and  no  variation  is  ever  gained.  Hence 
the  number  of  variations  lost  as  x  increases  from  a  value  a 
to  a  greater  value  j3  is  equal  to  the  number  of  real  roots  of  the 
equation  X  =  0  that  lie  betivcen  a  and  fi. 


STURM'S    THEOREM.  99 

149.  Scholium  I.  —  It  may  not  be  at  once  evident  to  the 
student  how  it  is  that  X,  which  (III)  is  of  the  same  sign  as 
Xx  immediately  after  x  has  increased  through  a  root  of 
X  =  0,  is  again  of  the  contrary  sign  before  x  increases 
through  the  next  root.  This  becomes  clear  upon  considering 
that  Xi,  being  the  first  derived  function  of  X,  must,  by 
Art.  82,  become  zero,  and  therefore  change  sign  for  some  value 
of  x  between  any  two  successive  roots  of  X  —  0.  Thus  Xx , 
which  was  of  the  same  sign  as  X  just  after  x  increased 
through  a  root  of  X  =  0,  will  be  of  the  contrary  sign  before 
x  increases  through  the  next  root  of  X  =  0.  By  IV,  this 
vanishing  of  X\ ,  or  of  any  of  the  functions  of  inferior  de- 
gree, does  not  alter  the  number  of  variations  of  sign  in  the 
series  of  functions,  but  merely  changes  the  distribution  of  the 
plus  and  minus  signs  in  the  series. 

150.  Sch.  2. —  If  for  some  value  of  x,  any  of  the  inferior 
functions,  as  Xr,  becomes  zero,  we  may  omit  that  function 
when  counting  the  variations ;  for  the  sign  of  the  vanishing 
function  must  form  a  permanence  with  the  sign  of  one  or 
other  of  the  adjacent  functions. 

151.  Cor.  1 . —  If  in  the  series  of  Sturm's  Functions  we 
successively  put  —  co  and  -f  co  for  x,  the  excess  of  the  num- 
ber of  variations  produced  by  the  first  substitution  over  the 
number  produced  by  the  second  is  equal  to  the  whole  number 
of  real  roots  in  the  equation. 

When  ±  co  is  put  for  x,  the  sign  of  a  function  is  always 
the  same  as  that  of  the  highest  power  of  x  in  that  function ; 
the  series  of  signs  for  x  =  +  co  is  therefore  the  same  as  the 
series  of  leading  signs  in  the  functions ;  the  series  of  signs  for 
x  —  —  &  may  be  easily  derived  from  the  preceding,  by  changing 
the  signs  of  the  leading  terms  that  contain  odd  powers  of  x. 

152.  Cor.  2.  —  In  order  that  an  equation  of  the  nth  degree 
may  have  all  its  roots  real,  it  is  necessary  and  sufficient  tit  at 
the  functions  be  n  +  1  in  number,  and  all  of  the  same  sign 
in  their  leading  terms. 

The  functions  will,  in  general,  be  n  -f-  1  in  number,  since 
each  remainder  is  usually  only  one  degree  lower  than  the  pre- 
ceding one.     If,  then,  there  be  n  -f-  1   functions,  all  having 


100  ALGEBRAICAL  EQUATIONS. 

the  same  leading  sign,  there  will  be  n  variations  of  sign  when 
x  —  —  </>,  and  no  variation  when  x  =  +  co  ;  there  are 
therefore  n  real  roots.  If  there  be  fewer  than  n  + 1  functions, 
then,  even  when  the  leading  signs  are  all  alike,  the  substitution 
of  —  co  for  x  will  give  fewer  than  n  changes  of  sign,  there  are 
therefore  fewer  than  n  real  roots. 

153.  Cor.  3.  —  If  the  functions  be  n  -f-  1  in  number,  but 
not  all  of  the  same  leading  sign,  there  are  as  many  pairs  of 
imaginary  roots  as  there  are  variations  in  the  leading  signs. 

For  suppose  there  are  m  variations  of  sign  in  the  leading 
terms,  then  for  x  =  +  co  there  will  be  m  variations  and 
n  —  m  permanences  of  sign.  For  x  =  —  co  the  foregoing 
variations  become  permanences,  and  the  permanences,  varia- 
tions ;  that  is,  for  x  =  —  co  there  are  n  —  m  variations.  The 
excess  of  the  number  of  variations  for  x  =  —  co  over  the 
number  for  x  =  +  co  is,  therefore,  n  —  2m ;  thus  there 
are  n  —  2m  real  roots  in  the  equation,  and  consequently  2m 
imaginary  roots. 

154.  Cor.  4.  —  The  number  of  positive  real  roots  is  equal 
to  the  excess  of  the  number  of  variations  in  the  signs  of  the 
final  terms  of  the  functions,  over  the  number  of  variations  of 
sign  in  the  leading  terms. 

For  when  x  =  0,  the  functions  reduce  to  their  final  terms 
and  present  as  many  variations  of  sign  as  these  do,  and  for 
x  =  -f  co  the  functions  have  the  same  signs  as  their  leading 
terms;  hence  the  number  of  variations  in  the  signs  of  the 
final  terms,  diminished  by  the  number  of  variations  in  the 
leading  terms,  represents  the  number  of  real  roots  between 
0  and  +  co. 

155.  Thus,  when  once  we  have  obtained  Sturm's  Functions, 
we  can,  by  mere  inspection  of  the  signs,  determine,  not  only 
how  many  of  the  roots  are  real  and  how  many  imaginary,  but 
also  how  many  of  the  former  are  positive  or  negative. 

To  ascertain  the  situation  of  each  of  the  positive  roots,  we 
successively  substitute  in  the  functions  the  series  of  numbers 

(',  1,  2,  3, &c,  till  we  reach  a  number  that  makes  the 

number  of  variations  of  sign  in  the  series  of  functions  equal 
to  the  number  of  variations  in  the  leading  terms.    Each  loss 


STURM'S    THEOREM.  101 

of  a  variation  in  the  signs  of  the  scries  of  functions  indicates 
the  passing  over  a  root.  If  more  than  one  root  be  found  to 
he  between  successive  integers,  we  may  proceed  to  substitute 
successive  fractions  till  we  separate  the  roots. 

The  situation  of  the  negative  roots  is  determined  in  a  simi- 
lar manner. 

156.  The  theorem  has  been  proved  on  the  supposition  that 
X  =  0  has  no  equal  roots.  We  shall  now  show  how  to  pro- 
ceed when  in  the  operation  for  finding  Sturm's  Functions  we 
obtain  a  zero  remainder,  showing  that  the  equation  has  equal 
roots. 

Let  the  equation  X  =  0  have  equal  roots,  then  X,  XY,  X2, 
and  the  functions  that  follow  will  have  the  last  function  Xp  as 
a  common  divisor.  We  can,  by  dividing  X  by  the  common 
divisor,  obtain  the  function  which,  equated  to  zero,  will  (94) 
contain  all  the  roots  without  repetition.  But  this  is  not 
necessary,  since  the  factor  Xp,  being  common  to  all  the  pre- 
ceding functions,  its  presence  or  absence  will  not  affect  the 
number  of  variations  of  sign  for  any  given  value  of  x.  For, 
if  Xp  is  positive  for  that  value  of  x9  its  presence  as  a  factor 
will  not  affect  the  series  of  signs  at  all,  while  if  Xp  is  negative 
it  will  reverse  all  the  signs.  Hence,  when  tve  discover  that 
X  =  0  has  equal  roots,  by  the  fact  of  some  remainder  becom- 
ing zero,  tve  can  employ  the  preceding  functions  to  determine 
the  number  and  situation  of  the  real  roots  that  are  unequal. 

The  common  factor  Xp  which  contains  the  remaining  roots 
may  be  analyzed  separately  to  ascertain  how  often  each  root  is 
repeated. 

157.  If  in  the  course  of  the  operation  for  obtaining  the 
functions  X2,  X3,  &c,  we  come  to  one  Xr,  which  we  know 
cannot  change  its  sign,  we  need  not  proceed  to  find  the  re- 
maining functions.  For  in  the  demonstration  of  the  theorem 
the  necessary  property  of  the  last  function  is  that  it  be 
incapable  of  becoming  zero.  As,  by  supposition,  Xr  cannot 
vanish,  the  demonstration  holds  good  for  the  functions  from 
X  to  Xr .  The  fact  that  a  function  cannot  change  sign  is 
most  readily  observed  in  the  case  of  a  quadratic  function, 
which  will  generally  be  the  second  from  the  final  remainder. 


102 


ALGEBRAICAL    EQUATIONS. 


As  the  labor  of  finding  Sturm's  Functions  increases  very 
rapidly  with  the  degree  of  the  equation,  and  especially  with 
the  last  functions,  it  is  of  some  importance  to  save,  if  possible, 
the  labor  of  computing  these  functions. 

Ex.  1.     Let  the  equation  proposed  for  analysis  be, 

X  =  xs  —  4a:2  —  11a?  +  43  =  0,     then 

X2  =  2x  ■ 
X2  =  9. 
By  Cor.  2,  the  roots  are  all  real,  and,  Cor.  4,  two  are  positive ; 
we  arrange  the  series  of  signs  for  different  values  of  x  as  in 
the  following  scheme : 


8*  —  11, 

7,     omitting  a  factor  40, 


X 

X 

Xx 

x2 

x3 

Variations 

0 

+ 

— 

— 

+ 

2 

1 

+ 

— 

— 

+ 

2 

2 

+ 

— 

— 

+ 

2 

3 

+ 

— 

— 

+ 

2 

4 

— 

+ 

+ 

+ 

1 

5 

+ 

+ 

+ 

+ 

0 

There  are  at  first  two  variations  of  sign,  and  the  number 
continues  the  same  till,  for  x  =  4,  we  have  one  variation  only : 
a  root  therefore  lies  between  3  and  4.  For  x  =  5  the  remain- 
ing variation  is  lost ;  thus  another  root  lies  between  4  and  5. 

It  will  be  sufficient  to  substitute  for  x  in  X  alone,  when 
but  one  root  remains  of  a  certain  sign,  as  a  change  of  sign  in 
the  final  term  will  show  when  the  root  is  passed.  Thus  to 
find  the  situation  of  the  negative  root  in  this  example,  we 
need  only  substitute  negative  values  for  x  in  X.  We  find  this 
root  to  lie  between  —  3  and  —  4. 


Ex.  2. 

Let  X  =  x5  —  5a?  +  10x2  —  20z  —  15  =  0,    then 

Xi  =  x*  —  ox2  -f-  4:X  —  4,  omitting  a  factor  5, 

X2  =  2x?  —  6r*  +  16«  +  15, 

X3  =  4.r2  +  55x  +  53, 

X4  =  -  3601a;  +  3431, 

X6=  -. 

STURM'S    THEOREM. 


103 


In  this  example  the  sign  only  of  the  final  function  is  deter- 
mined ;  for  this  function,  being  independent  of  x,  does  not 
change  sign.  As  in  this,  so  in  other  examples,  it  will  be 
sufficient  to  perform  as  much  of  the  numerical  work  as  will 
enable  us  to  determine  the  sign  of  the  final  remainder,  which 
sign,  when  changed,  will  be  the  constant  sign  of  the  final 
function. 

A  glance  at  the  leading  signs  informs  us,  Cor.  3,  that  the 
equation  has  two  imaginary  roots ;  by  Cor.  4,  there  is  but  one 
positive,  and  therefore  two  negative  roots.  Arranging  the 
series  of  signs  as  before,  we  have, 


X 

X 

xx 

X2 

Xs 

x4 

x5 

Variations 

-4 

— 

+ 

— 

— 

-f 

— 

4 

-3 

+ 

+ 

— 

— 

+ 

— 

3 

-2 

+ 

— 

— 

— 

+ 

— 

3 

-1 

+ 

— 

— 

+ 

+ 

— 

3 

0 

— 

— 

+ 

+ 

— 

— 

2 

1 

— 

— 

+ 

+ 

— 

— 

2 

2 

— 

+ 

+ 

+ 

— 

— 

2 

3 

+ 

4- 

+ 

+ 

— 

— 

1 

By  noting  where  the  losses  of  variations  occur,  we  see  that 
the  roots  are  situated  in  the  intervals,  (—4,  —3),  (—1,  0), 
(2,  3). 

Ex.  3.  Let  X  =  a?  -  60a;4 -f  10a;3-  120a;2  +  6x  -  1650  =  0, 
then  Xx  =  x5  —  40a*3  +  5a2  —  40a;  -f  1, 

X2  =  4a,4  —  x3  +  16a;2  —  x  +  330, 
X%  =  701a;3  —  68a;2  +  1959a;  +  314, 
X,  =  —  67288a;2  +  553067a;  —  40501906, 
11031642677a;  +  917865361528,       • 


X, 


X6=  +  . 

Here  we  see  by  inspection  of  the  leading  signs  that  there  are 
four  imaginary  roots,  and  of  the  real  roots  one  is  positive,  the 
other  negative.  We  can,  therefore,  readily  ascertain  the  situa- 
tion of  these  roots  from  X  alone.  The  roots  lie  in  the  inter- 
vals (-8,  -7),  (7,  8). 


104  ALGEBRAICAL  EQUATIONS. 

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STURM'S    THEOREM.  105 

159.  The  labor  of  computing  these  functions  is  evidently 
very  great,  and  that  of  substituting  different  values  of  x  in 
them,  in  order  to  ascertain  the  intervals  in  which  the  roots  lie, 
is  not  much  less,  even  when  we  content  ourselves  with  deter- 
mining between  what  integers  they  lie.  In  the  above  example 
the  three  roots  between  0  and  1  are  respectively  .366600.., 
.366660. .,  and  .366661. .,  all  three  concurring  to  four  places 
of  decimals,  and  two  of  them  to  five  places,  so  that  the  task 
of  separating  them  by  means  of  fractional  substitutions  in 
Sturm's  Functions  would  be  even  more  laborious  than  that  of 
obtaining  the  functions.  * 

Upon  the  whole  we  must  reach  the  conclusion  that  this 
celebrated  theorem,!  though  the  most  important  and  interest- 
ing addition  to  the  Theory  of  Equations  that  has  been  made 
in  the  course  of  the  past  two  centuries,  and  though  offering 
an  infallible  means  of  determining,  not  only  the  number  of 
real  roots  in  an  equation,  but  also  the  exact  situation  of  each, 
is  too  laborious  in  application  to  be  generally  available  for 
equations  of  above  the  fourth  degree. 

EXERCISES. 

Find  the  intervals  in  which  lie  the  real  roots  of  the  follow- 
ing equations  : 

1.  a?  -  6z2  +  7x  —  3  =  0. 

2.  x*  —  Sx*  +  ldx  —  11  =  0. 

3.  5s3  —  39z2  +  24z  —  25  =  0. 

4.  a*  —  5z?  +  3z2  +  Ix  +  2  =  0. 

5.  tf  —  &  _  x*  +  6  =  0, 

6.  x5  +  x3  —  2x*  +  Sx  —  2  =  0. 


*  It  will  be  shown  in  the  course  of  the  next  chapter,  in  connection  with  Horner's 
Method,  how  such  roots  may  he  readily  separated  without  the  aid  of  Sturm's 
Functions. 

t  The  theorem  was  discovered  by  M.  Sturm  in  1829 ;  received  the  mathematical 
prize  from  the  Academie  Royale  des  Sciences ;  was  published  in  Memoires  presentes 
par  des  Savans  Etrangers,  1835. 


10G  ALGEBRAICAL    EQUATIONS. 


CHAPTER    IX. 

HORNEE'S     METHOD.* 


160.  In  the  preceding  chapter  a  method  has  been  given 
by  which  we  can  determine  the  situation  of  each  real  root  in 
a  proposed  numerical  equation.  We  have  now  to  consider  a 
method  of  remarkable  simplicity  and  elegance,  by  which  we 
can,  after  determining  the  first  figure  of  a  real  root,  ascertain 
its  value  exactly,  if  commensurable,  or,  if  incommensurable, 
to  any  degree  of  approximation  desired. 

161.  Suppose  that  in  an  equation  f(x)  =  0  a  positive  real 
root  is  x  =  a  +  a'  -f-  a"  +  &c,  where  a,  a',  a",  &c,  are  its 
successive  figures  in  descending  order  of  the  decimal  scale, 
and  that  by  a  previous  analysis,  or  by  trial,  we  have  ascertained 
the  first  figure  a  of  this  root.  We  proceed  to  transform 
f(x)  =  0  into  another  equation  f  (a  +  x')  =  0,  whose  roots 
are  those  of  f(x)  =  0,  each  diminished  by  a.  We  now  pro- 
ceed to  ascertain  the  next  figure  a'  of  the  root  to  which  we 
are  approximating,  and  then  transform  /  (a  +  x')  =  0  into 
another  equation  whose  roots  are  each  less  by  a'.  Proceed- 
ing in  this  way,  we  determine  in  succession  the  figures 
a,  a',  a",  &c,  till  we  have  either  obtained  the  root  exactly, 
or  to  as  many  figures  as  are  necessary  to  the  degree  of  approxi- 
mation required. 

The  points  in  which  the  superiority  of  Homers  Method 
consists  are :  (1).  The  successive  transformed  equations  are 
obtained  from  the  preceding  equation  by  the  compact  and 
easy  process  already  given  in  Art.  69.  (2).  After  the  first 
figure  of  the  root  has  been  obtained,  we  are  guided  with  more 


*  This  method,  named  after  its  author,  Mr.  Horner,  of  Bath,  England,  was  pub- 
lished by  him  in  the  year  1819. 


HORNER'S    METHOD.  107 

or  less  exactitude  to  the  next  figure  by  a  process  which 
shall  presently  be  explained.  (3).  After  a  few  figures  of  the 
root  have  been  determined,  we  can  very  much  contract  the 
operation,  which  ultimately  becomes  a  case  of  contracted 
division. 

We  shall  now  consider  in  order,  I.  The  Method  of  Trans- 
formation;  II.  The  Method  of  Trial  Divisors;  III.  The 
Method  of  Contraction. 

162.  The  Method  of  Transformation.  —  This  has 
been  already  explained  and  illustrated  in  Art.  69 ;  we  apply 
it  as  follows : 

Let  f(x)  =  3a;3  —  1326a;2  -f  6576a;  —  9177  =  0,  and  sup- 
pose we  have  ascertained  that  a  root  lies  between  500  and  400 ; 
we  proceed  to  transform  f(x)  =  0  into  /(400  +  x')  =  0,  the 
roots  of  which,  are  those  of  f(x)  =  0,  each  diminished 
by  400. 


—1326 

+  6576 

-9177 

|  400 

1200 

—50400 

—17529600 

—126 

-43824 

-17538777  : 

=  /(* 

1200 

429600 

1074 

385776  = 

=  /i(400), 

1200 

2274  = 

:i/2(400). 

Hence  the  transformed  equation,  whose  roots  =  x  —  400,  is 

3a;'3  +  2274a;'2  +  385776a;'  -  17538777  =  0. 

We  could  now  proceed  to  find  by  trial  the  next  figure  of  the 
root  (400  4.  x'),  and,  when  it  is  found,  transform  by  the  figure 
thus  found;  but  we  shall  presently  show  how  wre  may  be 
guided  to  this  figure  without  unnecessary  trials. 

163.  It  is  important  to  remark  that  if  the  root  A,  to 
which  we  are  approximating,  is  the  only  positive  root  having  a 
for  its  leading  figure,  then,  in  the  transformed  equation,  A  —  a 
will  be  the  smallest  positive  root,  and  the  only  one  whose  lead- 
ing figure  is  in  that  order  of  the  decimal  scale  which  is  next 


108  ALGEBRAICAL  EQUATIONS. 

below  the  order  of  a.  For  each  root  whose  leading  figure  was 
less  than  a  will  be  negative  in  the  transformed  equation,  and 
all  the  roots  whose  leading  figure  was  greater  than  a  will  have 
their  leading  figures  of  higher  orders  than  a1,  the  leading  figure 
of  A  —  a.  Thus  supposing  the  roots  of  an  equation  to  be 
7,  34,  49,  56.3,  78,  290,  and  we  diminish  each  by  50,  then 
we  have,  —43,  —16,  —1,  6.3,  28,  240,  as  the  roots  in  the 
equation  transformed  by  50. 

If  we  again  diminish  each  of  these  roots  by  6,  we  obtain 
—  49,  —22,  —7,  .3,  22,  234,  as  the  roots  in  the  second 
transformed  equation.  Thus  the  root  towards  which  we  are 
approximating  becomes  rapidly  quite  small  as  compared  with 
the  other  roots.  Similar  remarks  apply  when  there  are  two  or 
more  roots  having  the  same  leading  figure. 

164.  Method  of  Trial  Divisors.  —  These  are  similar 
to  those  employed  in  the  evolution  of  arithmetical  roots,  and 
are  usually  deduced  as  follows : 

If  a  +  x'  is  a  root  of  f{x)  =  0,  and  x'  is  sufficiently  small 
as  compared  with  a,  then  —  -  ;  {  =  %'  approximatively. 

For,  by  Art.  69,  when  a  +  x'  is  put  for  x  in  f(x)  =  0, 
then 

f{a  +  x')  =  Cnx'»+  . .  .+ifi{a)x'*  +fl(a)x+f(a)  =  0. 

Now  if  x'  is  small  as  compared  with  a,  we  may  neglect  all 
the  terms  involving  higher  powers  of  x'  than  the  first ;   thus 

we  have  f\{a)x'  +  f(a)  =  0,  or  —  ^j-4  =  x'  approxi- 
matively. 


165.  The  proposition  is  generally  stated  as  above.  It 
would  be  better  to  state  the  second  condition  under  the  form, 
if  x'  is  small  compared  with  the  other  roots.  This  will  be 
more  clearly  seen  from  the  following  way  of  deducing  the 
rule  : 


HORNEK'S    METHOD.  109 

Let  a,  p,  y,  6, k,  and  x',  be  the  roots  of  the  trans- 
formed equation  f(a  +  x')  =  0,  then,  by  Art.  45,  we  have, 

—  f  (a)  =  aj3yd. .  .kx',   and 

fx  (a)  =  a(3yd. .  .k  +  x'(Qyd. .  .k  -f  ayd. . .«  +  &c.) 

f  (a)  afiyfi.  .  .kx'  , 

/i(#)    ""   a{fyd.  .K  +  x'((3yd.  .n-\-ayd.  .k-\-&c.) 
proximatively, 

when  x'  is  small  as  compared  with  the  other  roots. 

166.  The  degree  of  approximation  will  evidently  depend 
upon  how  small  x'  is  as  compared  with  the  remaining  roots, 
and  will  be  in  excess  over,  or  defect  from,  the  true  value  of  x', 
according  as  a(3yd. .  ,tc  and  x'((3yd. .  .k  -f  &c.)  have  contrary 
signs  or  the  same  sign. 

Now  in  163  we  saw  that,  if  a  +  x'  was  the  only  root  having 
a  certain  leading  figure,  then  x'  is,  even  in  the  first  trans- 
formed equation,  a  decimal  as  compared  with  the  other  posi- 
tive roots.  Of  the  negative  roots  some  may  be  numerically 
less  than  x'  in  the  first  transformation  (see  163),  but  at  the 
same  time  these  negative  roots  tend  to  diminish  the  expression 
x'((5yd. .  .k  +  ay 6. . .«  +  &c.)  by  making  the  products  vary 
in  sign. 

Thus  upon  the  whole  it  will  be  found  that,  even  in  the  first 
transformed  equation,  if  we  divide  minus  the  final  term  by  the 
preceding  coefficient,  the  figure  thus  obtained  will  not  be  far 
from  correct ;  in  many  instances  will  be  exactly  so.  In  the 
second  transformed  equation  the  trial  divisor  will  give  the 
next  figure  of  the  root  correctly,  with  comparatively  rare 
exceptions. 

If  the  figure  obtained  is  too  great,  we  shall  become  aware 
of  the  fact  before  completing  the  first  row  of  work  in  the  next 
transformation,  by  seeing  that  the  final  sign  will  be  changed, 
thus  showing  that  the  root  has  been  passed  over.  We  then 
try  the  next  lower  figure,  and  so  on  until  we  find  one  that  does 
not  cause  a  change  in  the  final  sign.  If  the  figure  obtained 
is  too  small,  we  shall  not  be  warned  of  the  fact  in  the  same 
way,  though  we  can  generally  infer,  from  the  result  produced 


110  ALGEBRAICAL    EQUATIONS. 

on  the  final  term,  whether  the  figure  would  require  to  be 
increased.  In  case  of  doubt,  it  would  be  prudent  to  try  in  a 
bye-operation,  whether  the  next  higher  figure  would  change 
the  final  sign,  which  it  ought  to  do  if  the  figure  obtained  by 
the  trial  divisor  is  correct.  Finally,  if  at  any  part  of  the  opera- 
tion we  transform  by  a  figure  too  small,  we  shall  be  informed 
of  the  fact  by  the  next  trial  divisor  giving  a  figure  of  too  high 
order  to  be  correct. 

167.  Eesuming  consideration  of  the  transformed  equation 
of  162, 

3a;'3  +  2274^2  +  385776a''  —  17538777  =  0, 

_ ■'  ,        /(400)  1753...         ,A   t  ,,       . 

we  find  that  —  J  x       .  ,  or  -  =  40  + ;  we  therefore 

transform  by  40  as  follows : 

3         +2274  +385776  -17538777     [40_ 

120  95760 

2394  481536 

We  need  proceed  no  further  with  this  transformation,  as  it 
is  obvious  that,  if  completed,  the  final  sign  will  be  changed. 
We  accordingly  transform  by  30,  thus : 


■2274 

+  385776 

-17538777  . 

|  30 

90 

70920 

13700880 

2364 

456696 

-3837897 

90 

73620 

2454 

530316 

90 

2544 
and  have  the  transformed  equation 

3x"s  +  2544a"*  +  530316a"  -  3837897  =  0. 

.  /(430)  3837...         „   ,  .        „         , 

As  —  ^nr^  j  or  -^k =  7  +  9   we  transform  by 

Ji  (4oU)  OoU ....  ** 

thus  : 


HORNER'S    METHOD. 


HI 


2544 

+  530316 

—3837897   |  7 

21 

17955 

3837897 

2565 

548271 

21 

18102 

2586 

566373 

21 

2607 

Hence  the  equation  has  a  root  437,  since  /(437)  =  0.  The 
remaining  roots  may  be  determined  from  the  depressed  equa- 
tion Sx'"z  +  26072;'"+  566373  =  0,  the  roots  of  which,  when 
increased  by  437,  are  roots  of  f(x)  =  0. 

For  the  sake  of  perspicuity  we  have  exhibited  each  trans- 
formation separately  in  the  preceding  example ;  in  practice 
we  proceed  as  follows  : 


3 

-1326 
1200 

+  6576 
—50400 

—9177  1 400  +  30  +  7 
—  17529600 

-126 
1200 

-43824 
429600 

-17^38777 
13700880 

1074 
1200 

385776 
70920 

—3837897 
3837897 

2274 
90 

456696 
73620 

2364 
90 

530316 
17955 

2454 
90 

548271 
18102 

3 

2544 
21 

566373 

2565 
21 

2586 
21 

3 

2607 

indicating  the  corresponding  terms  of  each  transformation  by 
a  small  index  figure,  or  other  mark. 


112  ALGEBRAICAL    EQUATIONS. 

168.   Ex.  2.     f{x)  =  x4  —  bz*  +  2z2  —  13a;  +  55  =   0 
has  a  root  lying  between  2  and  3  ;  we  proceed  as  follows  : 

-13 


-5 

+  2 

2 
-3 

—  6 
—4 

2 
—  1 

-2 
-6 

2 
1 

2 
i 

-4 

2 
i 

3 

.99 
—3.01 

.3 

1.08 

3.3 

—  1.93 

.3 

1.17 

21 
12 


+  55         |2.3 

-42 

-l 

13 

—10.1709 

2 

—33  2.8291 

—     .903 


■33.903 

•     ,579c 

-34.482' 


3.6  -    .76 

.3 


3.9 
.3, 

4.2' 


f  (2)         13 

In  the  first  transformation  we  found      /A    ,  or  —  =  .  3  + , 

/ i v~)  «*«* 

and  transformed  again  by  .  3,  as  above.     As  we  find     J  \   '' , 

2  8291  Ji^'0) 

or     *  =  .  08  + ,  we  could  proceed  to  transform  again  by 

.  08,  and  so  on  for  the  remaining  figures  of  the  root. 

169.  In  practice  it  is  found  convenient,  however,  to  dis- 
pense with  the  decimal  points.  This  we  effect  as  follows : 
When,  in  an  equation  of  the  nth  degree,  we  have  diminished  the 
root,  to  which  we  are  approximating,  till  the  next  figure  would 
be  a  decimal,  we  add  n  ciphers  to  the  right  hand  working 
column  when  completed,  (n  —  1)  ciphers  to  the  next  column 
when  completed,  and  so  on.  This  (59)  is  equivalent  to  multi- 
plying the  roots  each  by  10,  and  by  repeating  this  multiplica- 
tion at  every  transformation  we  are  enabled  to  dispense  with 
the  troublesome  decimal  points  in  the  working  columns,  the 
ciphers  also  serving  to  mark  where  a  column  was  completed. 


HORNER'S    METHOD.  113 

Beginning  the  example  again,  we  proceed  thus : 


-5 

+  2 
-6 
-4 

-13             +55     1 2.381 

2 
-3 

-  8            -42 
-21                130000 

2 
-1 

-2 
-6 

-12            -101709 
-33000            282910000 

2 

2 

-400 

-903         -276123264 

1 

-33903                67867360000 

2 

99 
-301 

-579             -34520621079 

30 

-34482000          333467389210000 

3 

108 

-33408 

33 

-193 

-34515408 

3 

117 
-7600 

-5504 

36 

-34520912000 

3 

39 

3424 
-4176 

290921 
-34520621079 

3 

3488 
-688 

295443 

420 

-34520325636000 

8 

3552 

428 

286400 

8 
436 

4521 

290921 

8 

4522 

444 

295443 

8 

4523 

299966C 

4520 
1 

10 

4521 
1 

4522 

1 

4523 
1 

45240 
In  every  instance  in  this  example  we  have  found  the  trial 
divisor  give  the  correct  figure  for  the  next  transformation,  and 


114  ALGEBRAICAL    EQUATIONS. 

we  could  go  on  in  the  same  way  to  find  as  many  more  figures 
of  the  root  as  may  be  desired.  But  when  we  examine  the 
second  working  column,  that  which  we  employ  as  trial  divisor, 
we  observe  that  the  addends  from  the  column  to  the  left  have 
less  and  less  influence  upon  the  leading  figures  of  this  column. 
It  is  evident,  indeed,  that  if  we  continue  to  perform  the  opera- 
tion without  contraction,  we  shall  soon  have  a  large  number 
of  superfluous  figures  in  the  second  and  right-hand  columns, 
which  figures  might  be  dispensed  with  without  affecting  the 
correctness  of  the  result  within  certain  limits.  We  therefore 
contract  the  work  in  the  manner  about  to  be  explained. 

170.  Method  of  Contraction. — When  we  have  obtained 
a  root  to  two  places  of  decimals  by  the  uncontracted  process,  ice 
can  contract  the  operation  by  ceasing  to  add  ciphers  to  the 
right-hand  column  when  completed,  by  striking  off  one  figure 
from  the  second  column  when  completed,  hvo  from  the  third, 
three  from  the  fourth,  and  so  on. 

This,  it  will  readily  be  seen,  is  equivalent  to  first  supposing 
the  coefficients  formed  as  in  the  preceding  article,  and  then 
dividing  each  by  10n  (n  being  the  degree  of  the  equation). 
Taking,  for  example,  the  coefficients  of  the  third  transforma- 
tion above, 

1     +  4520     +286400    -34520912000     +67867360000, 
and  dividing  each  by  10,000,  we  have 

.0001     +.452     +28.64     —3452091.2     +6786736. 

With  these  contracted  coefficients  we  proceed  as  before : 

|452      +28|64       -3452091|2         +6786736    1 196012 


4 

29 

—3452062 

|29 

—  3452062 
29 

-345203|3 
3 

3334674 
—3106800 

227874 
-207118 

—  345200 
3 

20756 
—20712 

-3|4|5|1|9|7 

44 
-35 

HORNER'S    METHOD.  115 

111  the  first  step  of   this  contracted  operation,  the   trial 

(678       \ 
'  * '  1  indicates  1  as  the  next  figure  of  the  root. 

Though  we  have  struck  off  all  the  figures  of  the  fourth  work- 
ing column,  yet  (as  in  contracted  multiplication)  we  allow  for 
its  influence  upon  the  third  column,  which  thus  becomes  29. 
We  complete  the  transformation  by  1,  and  strike  off  figures  as 
before.  The  trial  divisor  now  indicates  9  as  the  next  figure  ; 
we  take  no  further  notice  of  the  fourth  column,  but,  though 
we  have  struck  off  the  remaining  two  figures  of  the  third  col- 
umn, we  carry  3  from  it  to  the  second  column,  since  .29  x  9 
=  3  to  the  nearest  decimal.  We  then  complete  the  trans- 
formation, and  strike  off  another  figure  from  the  second  col- 
umn. The  rest  of  the  work  is  merely  a  contracted  division. 
The  approximated  root  thus  obtained  is  x  =  2.381966012. ., 
and  is  correct  to  the  last  figure,  which  would  be  1  if  obtained 
by  the  uncontracted  process. 

171.  Another  root  of  the  same  equation  lies  between  4  and 
5  ;  we  proceed  as  on  the  following  page  to  compute  this  root, 
contracting  from  the  third  transformation. 

Observe  here  that  the  sign  of  the  final  term  is  changed  in 
the  first  transformation.  This  arises  from  the  fact  that  a  root 
has  been  passed,  that  which  we  computed  in  170.  We  ob- 
serve also  that  the  trial  divisor  suggests  a  number  far  above 
the  true  value  of  x'.  This  arises  from  the  fact  that  x'  is  not, 
in  this  transformation,  much  smaller,  numerically,  than  the 
next  root. 

In  the  remaining  transformations  the  trial  divisor  invariably 
suggests  the  true  figure.     The  root  thus  obtained, 

x  =  4.61803399.., 

is  correct  to  the  nearest  decimal. 

172.  To  approximate  to  the  negative  roots  of  an  equation, 
we  change  the  signs  of  alternate  terms,  thus  (61)  changing 
negative  to  positive  roots,  and  proceed  as  in  the  above  exam- 
ples. The  roots  thus  obtained  will,  taken  with  the  negative 
sign,  be  the  negative  roots  of  the  equation. 


116 


ALGEBRAICAL  EQUATIONS. 


—5 

+  2 

—  13 

+  55   |4.618033996.. 

4 
—  1 

-4 

-2 

-  8 
-21 

—  84 
—290000 

4 

12 

40 

275856 

3 

10 

19000 

—  141440000 

4 

28 

26976 

77944941 

7 

3800 

45976 

—  63495059 

4 

696 

31368 

63224739 

110 

4496 

77344000 

—270320 

6 

732 

600941 

238542 

116 

5228 

77944941 

-31778 

6 

768 

602283 

23856 

122 

599600 

7854722|4 

-7922 

6 

1341 

48368 

7156 

128 

600941 

7903090 

—  766 

6 

1342 

48450 

716 

1340 

1 

602283 
1343 

79515|4|0 

2 

—50 

48 

1341 
1 

6036|26 
10 

79517 
2 

-2 

1342 

1 

6046 
10 

7|9|5jl|9 

1343 

6056 

1 

10 

1|344        60|66 


173.  By  this  compact  operation  we  are  thus  enabled,  when 
once  the  leading  figure  has  been  obtained,  to  approximate  to 
any  real  root  of  a  proposed  equation  with  just  the  same  amount 
of  numerical  labor  that  would  be  required  for  the  extraction 
of  an  arithmetical  root  of  an  order  corresponding  to  the  degree 
of  the  equation.  The  extraction  of  the  nth  root  of  a  given 
number  N  is,  in  fact,  nothing  else  than  the  solution  of  an 
equation  of  the  form  x11  —  N  =  0,  and  may  be  performed  by 
the  same  process,  and  with  the  same  contractions,  employed  in 
the  above  examples. 


HOKNER'S    METHOD.  117 

174.  We  have  thus  far  supposed  the  root  to  which  we  wish 
to  approximate  to  be  the  only  one  beginning  with  a  certain 
leading  figure.  We  can,  it  is  true,  by  means  of  Sturm's 
Method,  always  separate  real  roots,  however  closely  they  may 
coincide.  But,  as  shown  in  159,  the  labor  involved  in  doing  so 
is  excessive.  It  is  now  proposed  to  be  shown  that,  guided  by 
an  extension  of  the  rule  above  given  in  regard  to  trial  divi- 
sors, we  can  safely  proceed  in  the  approximation  by  Horner's 
Method,  even  when  there  are  in  the  given  interval  several 
equal  or  nearly  equal  roots. 

175.  It  has  been  shown  in  Art.  164,  et  seq.,  that  in  the 
transformed  equation 

f(a+x')  =  Cnx'»+  . . . .  if2  (a)**  +/i  («K+/(«)  =  0, 
when  x'  is  small  compared  with  a,  the  part  of  the  root 
already  found,  we  may  neglect  all  but  the  two  last  terms, 
A  (a)  +  /  («)  =  0,  from  which  simple  equation  we  may  ob- 
tain one  or  more  figures  of  x'.  Now  if  there  are  two  nearly 
equal  roots  in  the  interval  we  are  examining  in  f(a-\-x')  =  0, 
then,  since  (82),  the  first  derived  equation*  fL  (a  +  x')  =  0  has 
a  root  lying  between  them,  we  should  (164)  obtain  an  approxi- 
mation to  this  root  from  the  last  two  terms  of 

A  {a  +  x')  =  nCnX"-1  +  . . . .  l/3  {a)x'*+f2  {a)x'  +  fx  (a)  =  0, 

that  is,  —  A(a)  -^-/2{a)  =  x'  approximatively  when  there  are 
two  roots  having  figures  in  common. 

Generally,  if  f  (a  +  x')  =  0  has  r  equal  or  nearly  equal 
roots,  of  which  a  is  the  leading  part,  then  (82)  the  (r  —  l)th 
derived  equation  has  a  root  lying  between  the  two  least  of  these 
roots;  that  is, 

/,_!  (a+x')  = /,.  (a)x'  +  /_!  (a)  =  0 

has  a  root  a  little  greater  than  the  least  of  the  r  roots  of 
f{a  +  x')  =  0,  and,  by  Art.  164,  an  approximation  to  this  root 
is  given  by  the  expression  —  /;„i  (a)  -~  fr  (a).  Therefore, 
writing  the  transformed  equation  as 

Cnz'»+  ....  C'rx*+  C'r-,x'^+  ....  CV'3+  C'2x'*+  C\x' 
+  Co  =  0. 

*  By  the  r^  derived  equation  is  meant  the  r^  derived  function  equated  to  zero. 


118  ALGEBRAICAL    EQUATIONS. 

and  remembering  that  C'r  =  r—fr  (a),  we  have 

G  o  G  i  6  2  _  6  r_i  . 

~~C\'         2C2'         3CV rC"r> 

as  the  respective  expressions  for  suggesting  the  leading  figure 
of  the  next  transformation,  according  as  there  are  one,  two, 
three  or  r  roots  in  the  interval  under  consideration. 

We  deduce,  as  follows,  another  set  of  expressions,  which 
serve  as  a  check  upon  those  preceding. 

Let  f(a  +  x')  =  (p(xr)  i>(x'),  where  *l>(x')  is  the  product  of 
r  equal  factors  (x'—k)  and  <f)(xr)  the  product  of  the  n  —  r 
other  factors,  then  putting 

<p{x')    =  Cnx'"-r+ Dxx'     +  D0, 

ip(x')    =      x'r— rKx'r~\. .  ±_rKr~lx'  =F  xr, 

.-.  f(a  +  x')  =  Cnx'n+ ±{rD0«rl  -D^x'^D^  =  0, 

=  Cnx'»+ +  C\x'     +(7,0  =  0; 

G  n  -L'o  it  K  X  ,         , 

/. Try-  =  — ^ =r  =  — ,  or  — ,    nearly  when    n    is 

small  as  compared  with  the  other  roots. 

If  the  factors  in  ip(x')  are  not  absolutely,  but  yet  nearly, 
equal,  the  above  conclusion  will  still  hold;  thus  the  ex- 
pressions 


2<7'0  3<7'0  rC'o 


C\y  d' G\    ' 

ought  to  suggest  the  same  value  of  x'  as  the  corresponding 

expressions  in  A,  according  as  there  are  two,  three, or  r 

equal,  or  nearly,  roots  in  the  interval  under  examination. 

176.  When,  therefore,  we  have  ascertained  that  there  are 
several  roots  beginning  with  a  common  leading  figure,  we  pro- 
ceed as  follows : 

(1).  Transforming  in  the  usual  way,  we  diminish  the  roots 
by  this  common  figure.  \ 

(2).  By  means  of  the  proper  expression  in  A  we  determine 
the  next  figure.  If  the  expression  (A)  at  this  stage  suggests 
the  exact  figure,  that  from  (B)  will  usually  agree  with  it. 


HORNER'S    METHOD.  110 

(3).  We  continue,  at  each  transformation,  to  ascertain,  in 
the  same  way,  the  common  leading  figures  in  succession,  till 
either  a  discrepancy  in  the  figures  suggested  by  the  expressions 
(A)  and  (B)  warns  us  that  the  roots  no  longer  coincide,  or  a 
change  in  the  final  sign  shows  that  we  have  separated  a  root, 
which  will  be  the  least  of  those  to  which  we  were  approxi- 
mating. 

(4).  When  we  have  thus  separated  a  root,  or  group  of  roots, 
as  the  two  groups  do  not  differ  much  from  each  other  in  value, 
the  divisors  will  not  again  be  effective  till  we  have  ascertained 
the  correct  leading  figure  of  each  group  and  performed  a 
transformation  by  that  figure.  We  may  then  proceed  sepa- 
rately with  the  approximation  of  each  group,  guided  by  the 
expressions  in  (A)  and  (B)  appropriate  to  the  number  of  roots 
in  the  group,  till  we  separate  these  roots  also,  and  carry  on 
the  approximation  to  each  single  root  by  means  of  the  ordinary 
trial  divisor  as  far  as  desired. 

Ex.  1.     The  equation 

z4  —  437z2  +  3660  +  2826  =  0 

has  two  roots  between  12  and  13  ;  we  proceed  to  develop  them 
as  shown  on  the  next  page. 

2C'Q         4680       _, 
In  the  first  transformation  we  find -^r  ,  or  -^-  =  7 + , 

and  —  ^r  >  or  imT  =  8  +  '  we  transform  by  ?>  and  muI 
that  figure  correct.  In  the  next  transformation  we  find  both 
B4Q1C*2    and    _rfn26      suggest  0.8;  we  accordingly,  consider- 

ing  0  as  a  figure  of  the  root,  add  other  ciphers  to  the  working 
columns,  and  transform  by  8.     In  the  next  transformation  we 

begin  to  contract ;  both  ■  and  — — -  indicate  2  as  the 

next  figure,     In  the  next  transformation,  after  striking  off  the 

1 71  v  2  456 

contraction  figures,  we  find  -    indicates  0,  and    5Q     2 

indicates  4.  The  roots  therefore  coincide  no  further  ;  we  find 
that  1  produces  a  change  of  sign  in  the  final  term,  showing 


120 


ALGEBRAICAL    EQUATIONS. 


+  0 

2? 

12 
JL2 
24 

i? 

36 
12 

480 

_j? 

487 

_2 
494 
_7 
501 

7 


50800 


—467 
144 

—  323 

288 

-35 

432 


+  3660 

—3876 

—216 

-420 


+  2826 
—2592 


12.708203032 


2340000 
2339659 


39700 
3409 


—  636000 
301763 

—334237 
325969 


341000000CO 
—34070626304 


43109 
3458 

46567 
3507 


500740000 
406464 

501146464 
406528 

501552992 
406592 


5019595 
101 


5019697 
101 


5019799 
102 


—  8268000000 
4009171712 

—4258828288 
4012423936 

— 24640435|2 
10039395 

—  14601040 
10039598 


—45614 
1505 


4|2 
9 


■44108|5 
1506 


4260 

45 


4215 
45 


50|832       5|01|98|01 


•4|lj7;0 


29373696 

29202080 

] 71616 

-132325 

39291 

-37937 


1354 
-1251 


103 

—  84 
19 


that  the  figure  of  the  lesser  root  is  0.     We  carry  the  contrac- 

171 

tion  a  step  further,  and  by  the  ordinary  trial  expression  — — 

have  3  suggested  as  the  next  figure,  which  proves  to  be  correct. 
Carrying  on  the  approximation  for  a  few  steps  further,  the 
correct  figures  being  indicated  in  succession  by  the  trial  divi- 
sors, we  find  a  root  x  =  12.708203932,  which  is  correct  to 
the  last  figure. 

To  approximate  to  the  remainder  of  the  other  root,  we  take 
the  transformation  at  which  the  roots  separated,  and  find  by 


HORNER'S    METIIOD. 


121 


trial  that  8  is  the  greatest  figure  which  does  not  cause  a  loss 
of  two  variations  of  sign.    We  proceed  as  follows : 


5|01|98 


-456144J2 

4015841 

+  171616  |.0000869364 
-436482 

— 54560|2 
401584 

—264862 
226286 

34702 
3012 

,4 

—38576 
37059 

37714|4 
3012 

-1517 
1249 

4072 
45 

5 
1 

—268 
250 

4117| 
45 

7 

—  18 

4|1|6|3 

Adding  these  figures  to  those  already  obtained,  we  have  as  a 
second  root  x  =  12.7082869364..,  which  is  correct  to  the 
nearest  decimal. 

Ex.  2.     The  equation, 

W  +  391^  +  5376a;3  -  2344a;2  +  3S6x  —  16  =  0, 

has  three  roots  between  0'  and  1.  Before  proceeding  with  the 
solution,  as  on  the  two  following  pages,  we  multiply  the  roots 
by  10,  so  as  to  dispense  with  the  decimal  points.     We  find 

=  1+. 

707 


160X3  1     ,  A        22U  1     .  W      *  * 

OQa     =  1  +  ,  and  ■=— — -  =  1  +  .    We  transform  by  1,  and 
336  537x3  ^^  ' 

in  the  first  transformation  find  -  =  4-f,  and 


300  ~  '  7  55x3 

4  + ,  showing  that  the  three  roots  still  coincide.     In  the 


second  transformation  we  find 


729x3 


2  +,   and 


399 


938  '  '  55  x  3 

=  2  -f-,  which  figure  must  still  be  common  to  the  three  roots. 

882x3 


But  in  the  third  transformation  we  find 
631 


1404 


1  +  ,  while 


-  3  +,  showing  that  the  roots  no  longer  coincide. 

Upon  trying  1,  we  may  infer  from  the  effect  upon  the  final 
6 


122 


ALGEBRAICAL    EQUATIONS. 


7       +3910 

+ 537600 

—2344000 

3917 

3917 

541517 

3924 

541517 

-1802483 

3931 

3924 

545441 

3938 

545441 

-1257042 

39450  1 

3931 
549372 

549372 

i 

39478 

-707670000 

39506 

3938 

221955648 

39534 

55331000  * 

—485714352 

39562 

157912 

222587744 

395900  2 

55488912 

—263126608 

395914 

158024 

223220288 

395928 

55646936 

—39906320000 

395942 

158136 

11194247656 

395956 

55805072 

-28712072344 

39J5970  3 

3 

158248 

11195831368 

5596332000  * 

—17516240976 

791828 
5597123828 

11197415136 

—  63188258 

40 

791856 
5597915684 

5599539 

0 

-57588719 

4 

791884 
5598707568 

5599578 

6 

—51989140 

8 

791912 

5599618 

2 

5599499 

470* 

—463895 

22 

39 

6 

16798 

97 

5599539 

0 

—44709612 

39 

6 

16799 

5599578 

6 

—430297 

39 

6 

16799 

5 

559961812 

-4134 

98 

396 
515991658 

28 

99 

—4106 

98 

"5         4 

28 

—4078 

98 

28 

-40151 


IIORNER'S    METnOD. 


123 


+  3360000 
—  1802482 
"" 1557517 
-1257042 

3004750000  1 
—1942857408 
1061892592 
—1052506432 

93861600000 

—  57424144688 

36437455312 

—35032481942 


140497337 

—  57588719 


82908617 
-51989140 


3091947 
—1341288 


1750659 
1290891 


45976 
—20534 


25441 
20394 


-504 
243 

7 
1 

261 
242 

6 

—1600000 
1557517 


1421356 


—4248300000 

4247570368 


—  72963200000 
72874910624 


—88289376  * 
82908618 
-5380758  4 
5251977 
—  128781  5 
127210 
—  1571  6 
1570 


— 1 


19 


term  that  this  is  a  correct  figure  either  of  a  single  root,  or  of 
two  roots.     Upon  completing  the  transformation,  we  find  that 

the  1  is  a  figure  of  a  pair  of  roots,  for  =3+  and  also 

309 
— — -  =  3  4- .    In  the  next  transformations,  till  the  final  col- 
46x2                                         m                                       2C' 
umn  is  exhausted,  we  find  both  expressions, -^  and 

C\  c  1 

—  -jrjjr  ,  continue  to  agree  in  the  figures  indicated.     We  may 
2G  2 

therefore  suspect  that  the  roots  are  equal.     How  we  may 


124 


ALGEBRAICAL    EQUATIONS. 


co 

o 

«~ 

o 

CO 

o 

r. 

^ 

V 

co 

SO 

05 

X 

<Tv» 

X 

1 

io 

1 

CO  r-l 

IO  Tj< 

i-l  iO 

CO  "* 

GO  i-l 

?>-  t-i 

CO  CO 

CO  C5 

CO  o 

CO  CO 

CO  C5 

"*  o 

CO  CO 

C5  CO 

CO  CO 

CO  GO 

"*  co 

CO  tH 

^H  CO 

CO  IO 

iO    CO 

tH  OS 

T-l  C5 

GO   T-l 

CO  CO 

^  C5 

"*  CO 

H  Hi  1 

i-l  o 

O  Oi 

1 

1 

^  GO 

CO  i—l 

1 

1 

1 

1 

CO  CO 

CO  CO 

J>  2> 

O  i—l 

^  T-l 

©  J> 


+ 


o  eo 

o  CO 

CO 

CO  CO 

CO 

CO  ^ 

CO  CO 

CO 

o  CO 

o  o 

T-l   CO 

O  2> 

GO  CO 

co  io 
iO  o 

CO  o 

CO  CO 

T*   T-l 

iO  l— I 

CO 

co  © 

3>  O 

J>  o 

CO  o 

CO  CO 

i-l  CO 

©  GO 

2>  CO 

r-J  CO 

CO  CO 

•o  o 

CO  tH 

O  CO 

co  co 

CO 

CO 

^H 

o 

CO   T— 1 

^  s>- 

T— 1   CO 

O  iO 

iO  iO 

1— 1 

1— 1 

1—1 

1  T_l 

O  CO 

^  CO 

CO 

GO 

© 

© 

© 

© 

1  CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

r 

r. 

O 

30 

SO 

kO 

X 

CO 

X 

35 

— 

•-- 

'.-. 

^ 

~ 

^H 

co  ^ 

©  CO 
CO  CO 
£-  O 

©   T-l 

QO  O 

CO  CO 
CO  ^ 

1-1   "^ 

1 

2G411332  8 
44803596  0 

<3l 
CO  CO 

©  CO 

^  o 

T-l   O 

CO  00 

i—l  CO 

T-H  CO 

CO  CO 
iO  o 

T-l   O 

O  CO 
^  CO 
2> 

CO  CO 

CO  CO 

iO  o 
i-l  o 
GO  CO 
CO  GO 

o 

CO  CO 

i—l  © 
CO  CO 
© 
2> 

GO  GO 

o 

o  o 

^  CO 

o 

CO 

o  © 

O  CO 

o 

GO 

1-1  ?> 

CO  CO 

GO  2> 

GO  CO 

iO  i> 

©  CO 

t— 1  i—l 

CO   T-l 

"*  T-l 

CO  CO 

tH  CO 

"#  CO 

05 

o 

o 

© 

o 

o 

»o 

CO 

CO 

»o 

iO 

o 

HORNER'S    METHOD.  125 

verify  this  inference  will  be  hereafter  adverted  to  (187).  To 
compute  the  remaining  figures  of  the  third  root,  whose  leading 
figures  were  .142,  we  take  the  transformation  at  which  the 
roots  separated,  and  finding  by  trial  the  next  figure  8,  we  pro- 
ceed by  the  contracted  process  to  find  some  more  figures,  the 
trial  divisor  indicating  the  remaining  figures  correctly.  We 
thus  find  the  third  root,  x  =  .14285714. .. 

177.  These  examples  show  that  Horner's  Method  is  quite 
equal  to  the  separation  of,  and  approximation  to,  roots,  how- 
ever closely  coinciding  in  their  leading  figures.  Such  equal, 
or  nearly  equal,  roots,  which  have  usually  been  regarded  as 
presenting  great  difficulties  to  the  calculator,  may  really  be 
approximated  to  with  greater  facility  than  the  others,  since, 
guided  by  the  trial  expressions  A  and  B,  we  are  enabled  to 
calculate  a  part  of  several  roots  simultaneously  as  far  as  they 
coincide,  and  then  pursue  the  approximation  with  contracted 
coefficients  after  the  single  roots,  or  groups  of  roots,  diverge. 
For  practice  we  subjoin  a  few  equations,  each  having  a  pair  of 
roots  lying  between  0  and  1,  and  coinciding  for  several  figures. 
When,  as  in  these  examples,  there  are  only  two  such  roots  in 
a  given  interval,  the  separation  is  easily  effected  by  means  of 

26"  C ' 

the  trial  expressions -^rr  and  —  —^  ,  which  will  suggest 

the  same  figures  till  the  roots  separate.  At  that  point,  we  find 
by  trial  the  correct  leading  figure  for  each,  and  after  transform- 
ing, carry  on  the  approximation  to  each  by  means  of  the  ordi- 
nary trial  divisor. 

EXERCISES. 

+  152a;3  +  5101a;2  -  6780a;  +  2210  =  0. 
+  214a;3  +  10135a;2  —  8508a-  +  1770  =  0. 
+  284a;3  +  19171a;2  -  28530a;  +  10500  =  0. 
+  218a;3  +  10474a;2  —  17808a;  +  7440  =  0. 
+  188a;3  +  8651a;2  —  7900a;  +  1786  =  0. 
+  40a^  +  575a?  +  1588a;2  —  848a;  +  104  =  0. 
+  47a^  +  823a?  +  4964a;2  —  1454a;  +  104  =  0. 
+  63a;4  +  500a;3  +  1334a;2  —  729a;  +  91  =  0. 
+  93s4  +  233a;3  +  148a;2  —  300a;  +  80  =  0. 
x5  +  173a;4  +  2356a;3  +  10468a,2  —  14101a;  +  4183  =  0. 


1. 

& 

2, 

X* 

3. 

a;4 

4. 

7* 

5. 

a;4 

6. 

X5 

7. 

a;5 

8. 

X5 

9. 

X5 

10. 

X5 

120  ALGEBRAICAL    EQUATIONS. 


CHAPTER    X. 

ANALYSIS    OF    EQUATIONS    BY    FOURIER'S    THEOREM. 

178.  The  method  of  approximation  explained  in  the  pre- 
ceding chapter  requires,  as  a  necessary  preliminary,  that  the 
situations  of  the  roots  have  been  ascertained  by  a  previous 
analysis.  Sturm's  Method  is,  indeed,  fully  adequate  to  the 
analysis  of  any  numerical  equation,  but  is  at  the  same  time 
excessively  laborious  in  its  practical  application.  The  object 
of  the  present  chapter  is  to  show  how  the  necessary  informa- 
tion regarding  the  roots  may  be  obtained  independently  of 
Sturm's  Theorem. 

179.  Prop.  I. — Fourier's  Theorem.* 

Let  f(x)  =  0  be  an  equation  of  the  nth  degree,  then  if  a 
and  1),  of  'which  a  <  b,  be  successively  substituted  for  x  in 
the  series  of  functions  f(x),  f  (x),  fz{x), f  (x),  consist- 
ing of  f(x)  and  its  successive  derived  functions,  the  excess 
in  the  number  of  variations  of  sign  in  the  series  when  x  =  a. 
over  the  number  when  x  =  b,  will  not  be  less  than  the  number 
of  real  roots  of  f(x)  =  0  that  lie  between  a  and  b. 

By  Art.  8,  no  change  can  take  place  in  the  sign  of  any  of 
the  above  functions  except  when  x  increases  through  a  value 
that  makes  that  function  vanish. 

It  has  already  been  proved  (148,  III.)  that  if  a  function  f(x) 
vanish  when  x  =  c,  then  f(x)  and  its  first  derived  function 
f  (x)  have  contrary  signs  when  x  is  indefinitely  smaller  than 
c,  and  the  same  sign  when  x  is  indefinitely  greater  than  c. 
This  relation  must  evidently  hold  good  with  regard  to  f  (x) 


*  The  theorem  is  usually  named  after  Fourier,  though  first  published  by  Budan  in 
1807.  There  is  evidence,  however,  that  Fourier  had  developed  the  theorem  in  MS. 
as  early  as  1797. 


FOURIER'S    THEOREM.  127 

and  its  first  derived  function  f2(x),  and,  generally, with  regard 
to  any  function  fr  (x)  and  its  first  derived  function  /H  i  {x). 
We  have  two  cases  to  consider : 

I.  Suppose  that  for  some  value  between  a  and  b,  as  x  =  c, 

the  r  consecutive  functions  f(x),  fi(x),  f2(x), fr-i(%)> 

vanish,  but  f  (x)  does  not. 

From  what  has  been  said  above,  these  r  +  1  functions  will, 
when  x  is  indefinitely  smaller  than  c,  present  r  variations  of 
sign,  and,  when  x  is  indefinitely  greater  than  c,  no  variation. 

Hence,  as  x  increases  through  a  root  c,  that  occurs  r  times 
in  f(x)  =  0,  r  variations  of  sign  are  lost  in  the  series  of 
functions,  but  none  gamed. 

II.  Suppose  that  when  x  =  c  the  m  derived  functions 
f(x)  to  fr+m-i(x)  vanish,  but  f-i(x)  and  fr+m(x)  do  not. 

Then  in  the  series  of  m  +  1  functions  fr  (x), fr+m_i(x), 

fr+m(x),  there  will  be  m  variations  of  sign  when  x  is  in- 
definitely smaller  than  c,  and  f.  (x)  will  have  the  same  sign 
as  f.+m(x)  if  m  is  an  even  number,  and  the  contrary  sign 
if  m  is  an  odd  number.  Therefore  the  series  of  m  +  2 
functions, 

./Ufa),    f{x),  .  •  ..fr+m.X{x),   fr+m(x), 

will,  if  the  extreme  functions,  f-i(x)  and  fr+m(x),  have  the 
same  sign,  present  m  or  m  +  1  variations  of  sign  (according 
as  m  is  even  or  odd)  when  x  is  indefinitely  smaller  than  c, 
and  no  variation  when  x  is  indefinitely  greater  than  c ;  while 
if  fr-\(x)  and  fr+m(x)  have  contrary  signs,  there  will  be  m  +1 
or  m  variations  (according  as  m  is  even  or  odd),  when  x  is 
indefinitely  smaller  than  c,  and  one  variation,  when  x  is  in- 
definitely greater  than  c. 

Hence,  as  x  increases  through  a  value  that  makes  m  con- 
secutive derived  functions  vanish,  m  variations  are  lost  when 
m  is  an  even  number,  and  m  ±  1  when  m  is  an  odd  number ; 
that  is,  the  number  of  variations  lost  in  this  way  is  always  an 
even  number,  and  none  is  gained. 

Thus,  as  x  increases  from  a  to  b,  at  least  as  many  variations 
are  lost  (I.)  as  there  are  real  roots  in  the  interval ;  also  if  there 
are  more  variations  lost  than  there  are  real  roots  passed,  the 
excess  must  be  an  even  number  (II.). 


128  ALGEBRAICAL   EQUATIONS. 

180.  Cor.  1. — If  the  coefficients  of  an  equation  f(x)  =  0, 
present  m  more  variations  of  sign  than  those  of  the  trans- 
formed equation  f  (a  +  x')  =  0,  then  the  number  of  roots 
passed  in  the  transformation  is  m,  or  m  minus  some  even 
number. 

For  the  coefficients  of  f{x)  are  the  values  of  the  functions, 

g/.  M.  •  •  •  •  ]7/r  (*). ....  W  (*),  /i  H  /(*), 

when  0  is  put  for  a-,  and  the  coefficients  of  /(«  +  #')  are  the 
values  of  the  same  functions  when  a  is  put  for  x.  If,  then, 
there  are  m  variations  fewer  in  the  signs  of  the  coefficients 
of  f(a-\-x')  as  compared  with  those  of  f(x),  this  is  the  same 
as  a  loss  of  m  variations  in  the  signs  of  the  series  of  functions 

/„  (x)  . . . .  f.(x) /(#),   as  x  increases  from  0  to  a,  thus 

indicating  m  roots  in  the  interval,  or  m  minus  some  even 
number. 

181.  Cor.  2.  —  If  no  variation  is  lost  in  the  signs  of  the 
coefficients  of  a  transformed  equation,  as  compared  with  the 
preceding  one,  there  is  no  root  in  the  interval 

182.  Cor.  3.  —  If  any  odd  number  of  variations  is  lost, 
there  is  some  odd  number  of  roots  in  the  interval;  whether  more 
than  one,  we  cannot  say. 

183.  Cor.  4. —  If  any  even  number  of  variations  is  lost, 
there  is  either  no  root,  or  some  even  number  of  roots  in  the 
interval. 

184.  Cor.  5.  —  If  an  equation  have  an  odd  number,  p, 
of  consecutive  zero  coefficients,  these  zeros  indicate  p  ±  1 
imaginary  roots,  according  as  the  zeros  occur  between  similar 
or  contrary  signs  ;  if  there  be  an  even  number,  q,  of  consecu- 
tive zero  coefficients,  these,  ivhethcr  they  occur  between  similar 
or  contrary  signs,  indicate  q  imaginary  roots. 

This  follows  from  Art.  179,  II ;  for  if  we  imagine  the  pro- 
posed equation  transformed  into  another,  having  as  roots  those 
of  the  proposed  each  increased  by  an  indefinitely  small  quantity, 
the  zeros  would  be  replaced  by  numbers  having  alternately 
positive  and  negative  signs,  while  if  the  roots  were  diminished 


FOURIER'S    THEOREM.  129 

by  an  indefinitely  small  quantity,  the  zeros  would  be  replaced 
by  numbers  having  the  same  sign,  and,  as  by  supposition,  the 
zeros  occur  between  terms  that  are  not  zero,  variations  have 
been  lost  that  do  not  indicate  real  roots. 

185.  General  directions  for  the  application  of  this  theorem 
to  the  determination  of  the  intervals  in  which  positive  roots 
may  lie,  may  be  given  as  follows : 

(1).  Transform  successively  by  10,  100,  1000,  &c,  as  far  as 
necessary,  so  as  to  ascertain  how  many  roots  may  be  found  in 
the  intervals  [0,  10],  [10,  100],  &c.  This  step  may  in  many 
cases  be  omitted,  as  a  cursory  application  of  the  rules  for 
limits  (75,  76)  may  show  that  there  can  be  no  root  so  great 
as  10,  &c. 

(2).  Proceed  by  successive  transformations  by  1  to  find  be- 
tween what  units  the  roots  in  the  interval  [0,  10],  if  any,  lie, 
singly  or  in  groups ;  for  those  in  the  interval  [10,  100]  we 
proceed  by  10's;  for  those  in  the  interval  [100, 1000],  by  100's. 

(3).  Eoots  found  to  lie  singly  between  consecutive  units,  or 
tens,  or  hundreds,  may  be  regarded  as  ready  for  the  application 
of  Horner's  Method,  since  we  have  thus  the  leading  figure  of  a 
root  lying  singly.  But  if  more  than  one  root  is  indicated  be- 
tween consecutive  tens,  we  proceed  to  subdivide  the  interval, 
by  successive  transformations  by  1,  till  we  find  between  what 
successive  integers  the  roots  lie  singly  or  in  pairs ;  we  proceed 
similarly  when  several  roots  are  indicated  between  consecutive 
hundreds. 

186.  By  this  process  we  are  thus  enabled  either  to  obtain 
the  initial  figure  of  each  root  singly,  or  arrive  at  intervals  of 
consecutive  units  between  which  two  or  more  roots  may  lie. 
Proceeding,  in  this  case,  to  develop  the  roots  according  to  the 
precepts  given  in  Art.  176,  we  may  find :  (1)  by  their  separa- 
tion after  a  few  figures,  that  the  roots  in  question  are  real  and 
unequal  (see  examples  in  176) ;  (2)  by  their  continuing  to 
concur  to  seven  or  more  decimals,  we  may  infer  that  the  roots 
are  equal,  and  proceed  to  verify  the  inference  in  the  manner 
shown  below ;  (3)  the  roots  are  imaginary,  of  which  we  shall 
be  made  aware  by  tests  presently  to  be  explained. 


I     HKil 


130  ALGEBRAICAL    EQUATIONS. 

187.  We  shall  illustrate  each  of  these  cases,  taking  for  the 
first  example  the  difficult  equation,  already  given  in  Art.  158. 

Ex.  1.     x6  +  378Z5  +  38189^  +  4923682?  -  572554a* 

+  213720a;  —  26352  =  0. 

As  it  is  evident  by  inspection  that  there  is  a  root  less  than 
unity,  we  first  transform  by  1.  In  transforming  by  such  num- 
bers as  1,  10,  &c.,  there  is  no  effective  multiplication,  and  the 
easy  additions  and  subtractions  may  be  performed  mentally, 
the  results  only  being  set  down  as  in  the  following  example  : 

1  +378  +38189  +492368  —572554  +213720  —  26352  [_1_ 

379  +38568  +530936  —  41618  +172102  +145750' 

380  +38948  +569884 

We  need  pursue  this  transformation  no  further,  as  it  is  evi- 
dent that  all  the  three  variations  of  sign  in  the  proposed 
equation  will  be  lost ;  we  thus  know  that  there  is  at  least  one 
real  root  in  the  interval  [0,1],  possibly  three. 

To  ascertain  the  intervals  in  which  the  negative  roots  may 
lie,  we  change  the  signs  of  alternate  terms,  and  proceed  first  to 
transform  by  10. 

1-378  +  38189-492368-  572554-     213720-         26352  [10 
-368  +  34509-147278-2045334-20667060-206696952' 
-358  +  30929  + 162012-  425214-24919200' 
-348  +  27449  +  436502-3939806' 
-338  +  24069  +  667192' 
-328  +  20789' 
-318' 

As  the  number  of  variations  remains  the  same,  we  know 
that  there  is  no  root  in  the  interval  [0,  —10].  As  inspection 
of  the  transformed  coefficients  makes  it  probable  that  a  further 
transformation  by  10  will  cause  a  change  in  the  number  of 
variations,  we  proceed : 

1-318  +  20789  +  677192-3939806-24919200-206696952  [10 
-308  + 17709  +  854282  +  4603014  +  21100940  +     4412448 

We  need  proceed  no  further  with  this  transformation ;  one 
variation  is  lost,  and  it  is  plain  that  no  more  can  be  lost,  since 


FOURIER'S    THEOREM.  131 

the  second  term,  at  least,  will  remain  negative ;  a  root,  there- 
fore, lies  in  the  interval  [—10,  —20]. 

We  now  transform  by  100,  to  ascertain  whether  more  roots 
lie  in  the  interval  [0,  —100]. 

1  -378  +38189  -  492368  -      572554  -        213720-  23652  [100 

-278+10389+  546532+  54080646+5407850880  +  540785061648' 
-178-  7411  -  194568  +  34623846  +8870235480' 
-  78  -15211  -1715668  -136942954' 
+  22  -13011  -3016768' 
+  122-    811' 
+  222' 

Xo  variations  but  the  one  due  to  the  root  in  the  interval 
[—10,  —20]  are  lost  by  this  transformation;  no  roots,  there- 
fore, lie  in  the  interval  [—20,  —100]. 

We  proceed  to  a  further  transformation  by  100 : 

1  +  222  -     811  -3016768  -136942954  +  8870235480  +  540785061648  [J00 
322  +31389  +  122132-124729754-3602739920  +180511069648 
422  +73589  +7481032  +623373446  +  . . . . 

We  perceive  already  that  the  two  variations  will  be  lost  in 
this  transformation,  indicating  the  presence  of  two  roots  in  the 
interval  [—100,  —200].  We  can  now,  starting  from  the  trans- 
formation by  100,  transform  repeatedly  by  10,  or,  what  is  some- 
times preferable,  halve  the  interval  by  transforming  by  50. 
Proceeding  in  this  manner,  we  find  a  variation  is  lost  between 
the  transformations  by  160  and  170,  then  employing  only  the 
first  line  of  transformations,  we  easily  find  the  remaining  root 
is  situated  between  190  and  200.  We  have  thus  ascertained 
the  presence  of  single  real  roots  in  the  intervals  [—10,  —20], 
[-160,  -170],  [-190,  -200].  Of  the  three  roots  indicated 
in  the  interval  [0,  1]  we  know  that  one,  at  least,  must  be  real. 
As  these  roots  are  quite  small  as  compared  with  the  others,  we 
may  expect  the  trial  expressions  (^4)  and  (B)  for  three  roots 
(175)  to  suggest  the  same   leading  figures,  which  they  do, 

—37~  =  .3  +    and   49^  =  .3  +.      Proceeding  by  the 

aid  of  these  trial  divisors  to  develop  the  roots,  as  in  the  exam- 
ples in  Art.  176,  we  find  the  three  roots  concur  in  the  first 
four  figures ;  at  the  fourth  transformation  a  discrepancy  in  the 


132  ALGEBRAICAL    EQUATIONS. 

figures  suggested  by  the  two  trial  expressions  warns  us  that 
the  roots  no  longer  concur.  By  the  loss  of  one  variation, 
when  we  try  1  as  the  trial  figure,  we  find  the  root  separated  is 
the  smallest  of  the  three,  the  remaining  two  we  find  concur 
in  one  figure  more,  then  separate,  showing  that  they  too  are 
real. 


Ex.  2.     W  +  391^  +  5376z3  -  2344a?  +  336z  —  16  =  0. 

7     +  391     +  5376     _  2344    +   336—     16  [_1_ 
398     +5774     +3430    +3766  +  3750 

Before  the  completion  of  the  first  line  of  transformation  it 
is  apparent  that  all  three  variations  of  the  proposed  will  be  lost 
in  the  transformed  equation,  thus  indicating  three  roots  in  the 
interval  [0,  1].  To  ascertain  the  situations  of  the  remaining 
roots,  we  change  the  signs  of  alternate  terms,  and  proceed 
thus  : 


-391 

+  5376 

+   2344  +       336   +           16 

|10 

—321 

+  2166 

+  24004  +240376   +2403776 

—251 

—  344 

+  20564  +446016 

—181 

-2154 

—     976 

—111 

-3264 

—  41 

ariations  lost ;  we  transform  again  by  10 

-  41 

—  3264 

—    .976   +446016   +2403776 

|10 

+   29 

-2974 

—30716   +138856   +3792336 

+   99 

—1984 

—50556  — 3G6704 

+  169 

—  294 

—53496 

+  239 

+  2096 

+  309 

No  variations  lost ;  we  transform  again 

7   +309  +   2096  -  53496  -366704  +3792336  [10_ 

379  +  5886   +     5364  -313064  +   661696 

449  +10376  +109124  +778176 

519  +15566   +264784 

It  was  evident  before  the  completion  of  the  second  line  of 
this  transformation  that  all  the  coefficients  would  be  positive, 


FOURIER'S    THEOREM.  133 

the  loss  of  two  variations  indicating  two  roots  in  the  interval 
[—20,  —30].  But,  from  a  comparison  of  the  final  term  of  the 
last  transformation  with  that  of  the  preceding  one,  the  roots 
would  appear  to  lie  much  nearer  to  30  than  20.  It  was  there- 
fore worth  while  to  complete  the  third  line  of  the  transforma- 
tion so  as  to  obtain  the  coefficient  C'2 :    we  then  find  that 

~a\1,  or  ~tW~'  =  ~(1  +  )' and  also  i^tv  =  -(!+)' 

from  which  we  infer  that  30  is  greater  than  both  roots  by  1  +, 
i.  e.,  the  roots  lie  in  the  interval  [ — 28,  — 29],  which  inference 
is  verified  by  trial.*  Thus  the  roots  of  the  proposed  equation 
lie,  three  in  the  interval  [0,  1],  two  in  [ — 28,  — 29]. 

In  Art.  17G  the  three  roots  in  the  interval  [0,  1]  we  found 
to  concur  to  three  places  of  decimals,  at  the  fourth  figure  one 
separating.  The  remaining  two  were  found,  even  by  the  con- 
tracted process,  to  concur  to  seven  places  of  decimals,  a  fact 
suggestive  of  the  possibility  of  their  being  equal. 

Now,  by  Art.  95,  if  an  equation  have  two  incommensurable 
equal  roots,  it  must  have  another  pair,  and  be  divisible  by  a 
quadratic  factor  having  commensurable  coefficients.  If,  there- 
fore, as  in  this  example,  besides  the  pair  of  roots  supposed  to 
be  equal,  another  pair  of  roots  is  indicated  in  another  interval, 
these,  if  the  roots  are  really  equal,  must  be,  what  may  be 
termed,  the  conjugates  of  the  supposed  equal  roots,  and  have 
their  decimal  parts  the  same,  if  the  two  pairs  are  of  opposite 
signs,  or  complementary,  if  of  the  same  sign ;  also  the  product 
of  a  conjugate  pair  must  be  integral.  Here  we  have  the  sup- 
posed equal  roots  each  =  .  1421356 . . ;  the  other  pair,  if  the 
supposition  is  correct,  must  each  =  —28.1421356..;  the 
sum  of  a  pair  of  these,  with  changed  sign,  is  28,  their  product 
is  —  4  approximatively.  We  try  whether  x2  +  28a?  —  4  will 
divide  the  first  member  of  the  equation ;  as  it  does  so,  we  know 
that  there  are  two  pairs  of  equal  roots. 


*  Whonevor,  in  the  course  of  our  search,  we  find  a  root,  or  roots,  indicated  in  a 
certain  interval,  we  may,  as  in  this  example,  employ  the  coefficients  of  that  trans- 
formation which  has  the  smaller  final  term  to  guide  us  to  the  next  figure,  the  sign  of 
the  suggested  figure  showing  whether  it  is  to  he  added  to  the  smaller,  or  subtracted 
from  the  greater,  of  the  numbers  between  which  the  roots  are  indicated.  Even  when 
the  final  terms  show  no  decided  difference,  the  numbers  suggested  will  often  be  com- 
plementary, or  nearly  so,  thus  showing  how  far  we  may  depend  on  them. 


134  ALGEBRAICAL  EQUATIONS. 

Similarly,  if  r  roots  a  +  S,  where  6  is  the  decimal  part,  are 
found  to  concur  to  so  many  places  of  decimals  as  to  render  it 
probable  that  they  are  equal,  and  there  is  another  interval  con- 
taining 5  possibly  equal  roots  with  an  integral  part  b,  then  if 
(a  +  <5)  (#  +  <5)  =  c,  an  integer,  approximative^,  there  is  prima 
facie  evidence  that  some  of  the  roots  are  equal,  in  which  case 
x2  —  (a-\-b)x  +  c*  will  divide  the  first  member  of  the  equation 
as  many  times  as  there  are  pairs  of  conjugate  roots. 

Ex.  3.     x5  —  x*  —  15a;3  —  hx2  +  53z  +  51  =  0. 

We  proceed  as  in  the  preceding  examples  to  ascertain  the 
interval  in  which  roots  may  lie ;  as  the  method  of  procedure 
has  been  sufficiently  exemplified,  we  shall  give  merely  the 
results. 


[0]. 

1 

—  1 

-  15 

—     5 

+   53 

+  51, 

two  variations. 

[1]. 

1 

+   4 

—     9 

-  46 

—     1 

+  84, 

(6              a 

ra- 

1 

+   9 

+  17 

—  39 

—  99 

+  33, 

a              a 

[3]. 

1 

+  14 

+   63 

+  76 

-  85 

-78, 

one  variation. 

w. 

1 

+  19 

+  129 

4-359 

+  317 

-  9, 

a             a 

It  is  evident  that  the  remaining  variation  will  disappear  in 
the  next  transformation  ;  the  positive  roots  are  therefore  real, 
and  lie  in  the  intervals  [2,  3],  [4,  5].  To  find  the  places  of 
the  negative  roots,  we  change  the  signs  of  alternate  terms,  and 
proceeding  in  the  usual  way,  obtain  the  followiug  results  : 

[0].     1     +1     —15     +5     +53     —51,    three  variations. 

[1].     1+6—1     —24     +27     —  6,       " 

[2].     1     +11     +33     +19     +5     +3,    no  variation. 

AVe  infer  from  the  loss  of  three  variations  that  there  is,  at 
least,  one  real  root  in  the  interval  [— 1,  —2],  possibly  three. 
The  fact  that  two  of  the  roots  are  imaginary  is  shown  at  once 
by  the  test  we  are  about  to  establish. 

188.  Def.  —  In  Art.  66  it  was  shown  that  if  Cr  be  the 
coefficient  of  xr  in  an  equation  f(x)  =  0,  then 

C2  -  2  Cr+l  •  CU  +  2  Cr+2  •  CU  -  ^  Ch-3 '  CU  +  &c, 
will  [taken  negatively  if  xr  occur  in  an  even  term  in  /(#)] 

*  Or  x"*  —  (a  +  b  +  Dec  +  c,  when  a  and  b  have  like  signs. 


TESTS    FOR    IMAGINARY    ROOTS.  135 

be  the  coefficient  of  (x2)r  in  F(x2)  =  0,  the  equation  that  has 
for  roots  the  squares  of  the  roots  of  f(x)  =  0  :  this  expres- 
sion, or  its  numerical  value,  we  shall  refer  to  as  a  coefficient 
fa  net ion  of  f(x). 

189.  Prop.  II. —  There  are  at  least  as  many  imaginary 
roots  in  an  equation  f(x)  =  0  as  there  are  variations  in  the 
signs  of  its  coefficient  functions. 

For,  since  F(x2)  =  0  must  have  as  many  positive  real  roots 
as  f(x)  =  0  has  real  roots,  the  number  of  real  roots  in 
/  (.r)  =  0  cannot  be  greater  than  the  number  of  variations  of 
sign  in  F(x2),  and  consequently  the  number  of  imaginary 
roots  cannot  be  less  than  the  number  of  permanences  of  sign 
in  F(x2).  Now  of  the  coefficient  functions  the  first  C2  and. 
the  last  C02  are  always  positive,  so  that  the  number  of  varia- 
tions in  the  signs  of  the  whole  series  of  functions  must  be  an 
even  number,  if  any,  and  indicate  an  equal  number  of  perma- 
nences in  the  signs  of  F(x2),  since  the  values  of  these  expres- 
sions, with  the  sign  of  each  alternate  one  changed,  are  the 
coefficients  of  F(x2) ;  the  number  of  imaginary  roots  in 
f(x)  =  0  cannot,  therefore,  be  less  than  the  number  of  varia- 
tions in  the  signs  of  the  coefficient  functions. 

190.  Cor.  —  If  the  square  of  any  coefficient  Cr  be  less  than 
Cr+i'Cr.i,  the  product  of  the  contiguous  coefficients,  a  pair  of 
imaginary  roots  is  thus  indicated. 

If  f(x)  =  C„xn+  . . .  Cr+lxr+l+Crxr+Cr-lxr-1+. . .  C0  =  0, 
then  the  (r—l)th  derived  function,  when  divided  by  \r— 1, 

will  have  for  its  last  three  terms   -±— -— lCr+lx2+  rC,x  +  <?,._, ; 

)& 

there  will  be  a  pair  of  imaginary  roots  in  fr-i(x)  =  0,  and 
therefore  (88),  in  f(x)  =  0,  if  r2C2  <  r{r+l)Cr+l-Cr-i,  or 

C2<  r±l<7"  .(*4    and,  a  fortiori,  if  C2  <  Cr+1'Cr^. 
r 

This  corollary  will  often  be  found  sufficient  to  determine  the 

character  of  the  roots  in  a  doubtful  interval ;  in  a  nice  case  it 

r-fl 
may  be  important  not  to  neglect  the  factor   ,  especially 

when  r  is  a  small  number. 


136  ALGEBKAICAL    EQUATION'S. 

191.  The  test  provided  by  the  above  proposition  is  so 
simple  in  its  application  that  a  glance  at  the  coefficients  of  an 
equation  will  often  suffice  to  show  that  some,  if  not  all,  of  its 
roots  are  imaginary. 

Ex.  1.    x5  +  z4  +  &  —  2z2  +  2z  —  1  =  0. 

Here  the  coefficient  functions,  as  found  by  the  rule,  are, 
1     — 1     +9     —  2     ±0     +1,    four  variations ; 
there  are  accordingly  four  imaginary  roots  in  the  equation. 

It  is  obviously  not  necessary  to  obtain  the  actual  values  as 
we  have  done  in  this  example,  the  signs  alone  being  important. 
The  first  and  the  last  are  always  positive,  and  the  rest  are 
usually  obvious  to  inspection. 

Ex.  2.     x1  —  2x5  —  Sx3  +  4z2  —  5x  +  6  =  0. 

When,  as  in  this  example,  the  equation  is  not  complete,  we 
must  supply  the  place  of  the  absent  terms  by  zeros  before 
proceeding  to  determine  the  signs  of  the  coefficient  functions. 

We  here  obtain  the 'series  of  signs,  +  H f-  H h> 

and  see  that  there  are  at  least  four  imaginary  roots. 

Ex.  3.    5xG  -  ldx5  +  43z*  —  75z3  +  438^-412^  +  253  =  0. 

From  this  we  obtain  the  series  of  signs,  H 1 1 \-, 

and  see  that  the  roots  are  all  imaginary. 

In  the  last  example  of  Art.  187  we  arrived  at  a  transforma- 
tion, 

1     +11     +33     +19     +5     +3, 

where  two  variations  had  been  lost,  the  corollary  above  given 
shows  that  the  two  roots  in  the  interval  are  imaginary  since 
52  <  19  x  3,  thus  completing  the  analysis  of  the  equation. 
Having  showed  that  variations  in  the  signs  of  the  coefficient 
functions  indicate  imaginary  roots,  we  have  now  to  prove  the 
converse  of  that  proposition. 

192.  Prop.  III.  —  If  an  equation  f(x)  =:  0  have  a  pair 
of  imaginary  roots  a  ±  j3\/—  1,  there  are  certain  limit* 
/I  and  n,  one  less  than  a,  the  other  greater,  such  that,  if  the 
equation  be  transformed  to  f(y-\-x')—{),  where  y  is  any 


TESTS    FOR    IMAGINARY    ROOTS.  137 

real  quantity  between  X  and  [*,*  one  at  least  of  the  coefficient 
functions  icill  be  negative. 

(1).  If  f{x)  =  0  be  transformed  into  /(«  +  #')  =  0,  in  the 
latter  equation  the  roots  a  ±  [3  V—  1  will  be  reduced  to 
±j3\/ — 1,  which  roots  in  F\a-\-x'j  =  0  will  become  a 
pair  of  negative  real  roots,  —  (3%  —  /32.  These  (53)  must  pro- 
duce a  pair  of  permanences  in  the  signs  of  the  coefficients  of 
F(a-\- x'z),  corresponding  to  which  must  be  a  pair  of  varia- 
tions in  the  signs  of  the  coefficient  functions  of  f(a  +  x').  Hence 
there  is  a  transformed  equation  which  has  at  least  one  negative 
function  corresponding  to  the  supposed  pair  of  imaginary  roots. 

(2).  Supposing  f(x)  to  be  of  the  nth  degree,  then 
Forming  the  coefficient  functions  according  to  188,  we  have 


-sfttjiUjr) ;  ■  •  •  •  UAy)J-My)f(y) ; 


U(y)Y- 

Of  these  functions,  which  are  all  of  even  degree,  the  first 
and  last  cannot  be  negative,  and  of  the  rest  none  can  become 
negative  if  all  the  roots  of  f(x)  =  0  are  real  (189).  But, 
if  f(x)  =  0  have  a  pair  of  imaginary  roots  a  ±  (3  V—  h 
it  has  been  proved  that  some  one  of  these  functions,  say  [/i(y)]2 
—  fz(y)f(y),  becomes  negative,  at  least  when  y  =  a.  This 
function  must  therefore  become  zero  for  some  value  of  y 
greater  than  a,  as  \i,  and  also,  being  of  even  degree,  for  some 
value  A,  less  than  a.  Thus,  if  the  roots  of  f(x)  =0  be 
diminished  by  any  quantity  between  X  and  \x,  at  least  one  of 
the  series  of  coefficient  functions  in  the  transformed  equation 
will  be  negative. 

193.  We  say,  at  least  one,  for  several  consecutive  functions 
may  be  negative  for  the  same  value  of  y,  and  these,  in  like 
manner,  continue  negative  for  values  of  y  between  other  limits 

*  The  signs  of  these  limits  may  of  course  he  different,  and  one  of  the  admissible 
values  of  y  he  0,  as  in  the  examples  in  Art.  191. 


138  ALGEBRAICAL    EQUATIONS. 

[A',  jit'],  &c.  If  these  negative  functions  were  not  consecutive, 
there  would  be  more  than  one  pair  of  variations,  and  therefore 
more  than  one  pair  of  imaginary  roots  indicated. 

194.  By  considering  the  subject  in  the  following  manner, 
we  obtain  some  insight  into  the  relations  existing  between  the 
limits  [A,  fi]  and  the  roots  a  ±  0  V—  1. 

Corresponding  to  a  pair  of  roots  a  ±  j3  V—  1  in  f(x)  =  0 
is  a  factor  x2  —  lax  +  (a2  -f  /32)  in  f(x),  and  a  factor 
&  _  2  (a2  _  (32)x2  +  (a2  +  02)2  in  F(x2). 

As  long  as  a'2  >  (32  (where  a'  =  a  —  y),  the  real  part  of  the 


imaginary  roots  (a'±(3y  —  l)2  in  F(y  +  x'  )  =  0  remains 
positive;  but,  since  as  y  increases  in  f(y -\-x')  =  0,  a'  be- 
comes smaller  while  (3  remains  constant,  a'2—l32  first  becomes 
negative ;  and,  when  y  =  x,  the  factor  in  F(y  +  x'2)  =  0 
becomes  x*  -f  2(322;2  +  j34,  representing,  as  we  have  seen,  a  pair 
of  real  negative  roots.  As  y  still  increases,  a'2,  after  passing 
through  zero,  increases  till  it  again  reaches  a  value  at  which 
the  coefficient  —  2  (a'2—  (32)  no  longer  causes  permanences  in 
the  signs  of  F(y-\-x'2).  Thus,  as  —  2  (a'2  —  ft2)  is  positive 
only  when  a'  has  a  value  between  +  (3  and  —  (3,  it  would 
appear  that  the  difference  between  A  and  \i  must  generally 
be  somewhat  less  than  20,  or  A  and  \i  lie  between  a  —  (3 
and  a  +  (3. 

Or,  suppose  F(y+~x~'2)  =  <p{z2)[z*-2(a'2-(32)z2  +  (a'2  +  (32)2,] 
where  z  =  y  +  #',  and  $(22)  is  the  function  involving  the 
squares  of  the  real  roots,  and  of  those  imaginary  roots  within 
whose  limits  we  shall  first  suppose  y  does  not  here  fall.  The 
signs  of  (p(z2)  will  present  a  series  of  variations,  say, 

+  -  +  -  +  -  +  -+. 
Now,  if  in  the  factor  z*  -  2  {a'2  -  (32)z2  +  (a'2  +  /32)2,  we 
have  a'2  =  (32,  the  series  of  signs  in  that  factor  will  be 
4-  0  +,  and  the  multiplication  by  that  factor  will  merely 
add  two  variations  to  those  of  (p(z2),  since  like  signs  will  come 
under  like  :  thus  in  this  case  y  =  a  ±  (3  does  not  fall  within 
the  limits  of  a-±(3\/—  1.  But  if  the  signs  of  <f)(z2)  do 
present  a  pair  of  permanences,  let  the  series  of  signs  be, 

+    -    +    -    +    -    +    +    +, 


FOURIER'S    THEOREM.  139 

then  the  multiplication  by  the  factor  presenting  the  series  of 

signs  +  0  -f  will  produce  the  series  H 1 1 h  ±  +  +  +  > 

which  may  have  four  permanences:  thus  y  =  a  ±/3  may  fall 
within  the  limits  of  the  roots  a  ±  j3a/ — 1,  if  at  the  same 
time  it  be  within  the  limits  of  another  pair  of  imaginary 
roots. 

195.  Hence,  if  there  be  two  pairs  of  imaginary  roots, 
a  ±  fiV  —  1  and  y  ±  6  V—  1,  then  of  the  four  quantities, 
a-ft  a  +  ft  y-d,  y  -f  d,  (1)  if  a  +  [3  =  or  <  y-d,  the 
two  pairs  of  roots  will  indicate  their  presence  by  producing 
pairs  of  variations  in  the  signs  of  the  coefficient  functions  of 
different  transformations  :  (2)  if  both  a  —  (3  and  a-f/3  lie 
between  y  —  6  and  y  -f-  6,  the  two  pairs  of  roots  will  indicate 
their  presence  by  four  variations  in  the  coefficient  functions 
of  some  one  transformation ;  this  includes  the  cases  where 
a  =  y,  whether  (3  =  d  or  not :  (3)  if  a  -f-  j3  lies  be- 
tween y  —  (5  and  y  +  d,  the  two  pairs  of  roots  may  or 
may  not  give  indication  of  their  presence  in  the  same  trans- 
formation. 

196.  If  the  difference  between  X  and  \i  be  not  less  than 
unity,  whenever,  indeed,  A  and  \i  do  not  both  lie  between 
consecutive  integers,  the  character  of  imaginary  roots  is  easily 
determined.  For  when  in  the  course  of  our  transformations 
to  discover  in  what  intervals  roots  may  lie,  we  have  narrowed 
down  a  doubtful  interval  to  between  two  consecutive  integers, 
one  or  other  of  these  integers  must,  in  the  supposed  case,  fall 
between  the  limits,  and  one  or  other  of  the  transformations 
have  a  negative  coefficient  function.*  We  shall  first  illustrate 
this,  the  more  common  case,  by  a  few  examples,  leaving  the 
more  difficult  cases  for  future  consideration. 


*  The  ftinction  If  Ay)]2  -fMfiy)  equated  to  zero,  or  <p(y)  =  0,  of  which  X  and  ft 
are  roots,  may  he  regarded  as  a  limiting  equation  to  the  real  parts  of  imaginary  roots 
of  fix)  =  0,  just  as/^aj)  =  0  is  to  the  real  roots.  The  first  derived  function  of 
<p{y)  equated  to  zero,  or  My)f,{y)  -f3(y)f(y)  =  0,  must  have  a  real  root  between 
X  and  /i.  We  could,  therefore,  if  there  were  not  easier  methods,  always  thus  deter- 
mine a  value  of  y  which  would  make  f(y  +  x')  present  variations  in  the  signs  of  its 
coefficient  functions,  if  the  roots  in  the  doubtful  interval  are  imaginary. 


140  ALGEBRAICAL    EQUATIONS. 

Ex.  1.    Let  the  equation  proposed  for  analysis  be, 

4Z5  —  28a4  —  39a8  +  504z2  —  239a;  —  1855  =  0. 

We  see  that  there  is  probably  no  root  greater  than  10 ; 
we  therefore,  transforming  successively  by  1,  obtain  the  follow- 
ing results : 


[0]. 

4  —28 

[!]• 

4-8 

[•>]. 

4  +12 

ra- 

4 +32 

te 

4  +52 

39   +504  -239  —1855 

111   +259  +560  —1653 

103  —  82  +733  —  949 

15  -279  +328  -  385 

153  _  92  —127  -  315 


three  variations. 


one  variation. 


Here  two  variations  are  lost,  and  since  obviously  12  7 2 
<  184  x  315,  the  roots  indicated  are  imaginary.  The  re- 
maining positive  root  must  be  real,  and  its  interval  [5,  6]  is 
found  by  finding  what  number  will  make  the  final  term 
positive. 

To  find  the  intervals  of  the  negative  roots,  we  change  the 
signs  of  alternate  terms,  and  transforming  successively  by  1, 
obtain  the  following  results  : 

[0].     4+28  —  39  —504  —  239   +1855;    two  variations. 

[1].     4  +48   +113  —413  —1232   +1105;      " 

[2].     4  +68   +345   +254  -1507  -  375;    one  variation. 

The  loss  of  a  variation  here  shows  that  the  negative  roots 
are  real,  and  one  lies  in  the  interval  [—1,  —2].  The  remain- 
ing root  is  found,  as  above,  to  lie  in  the  interval  [—3,  — 4]. 
The  proposed  equation  has  therefore  three  real  roots  lying 
in  the  intervals  [—3,-4],  [  —  1,-2],  and  [5,  6],  and  two 
imaginary  roots,  the  real  part  of  which  is  not  far  from  4. 

Ex.  2.     Let  the  equation  proposed  for  analysis  be, 
xg  _  6%5  +  40a?  +  (3(^2  _  x  _  i  _  0. 

Proceeding  as  above,  we  obtain  the  results  : 

[0].  1  —  6  +  0  +40  +    60  —     1  -     1;  three  var. 

[1].  1  +   0  -15  +   0   +135   +215  +   93;  two  var. 

[v].  i  +   6  +   0  —40   +   60   +431  +429;  "       " 

[3].  1  +12  +45  +40  +  15  +467  +887;  no  var. 


FOURIER'S    THEOREM.  141 

The  loss  of  a  variation  indicated  a  real  root  in  the  interval 
[0,  1] ;  the  loss  of  two  variations  in  the  interval  [2,  3]  leads 
us  to  apply  the  test  for  imaginary  roots,  and  as,  obviously, 
15  <  40  x  407,  the  roots  indicated  are  imaginary.  To  find 
the  negative  roots,  changing  alternate  signs,  we  have  : 

[0].     1   +   6   +   0  —40   +G0   +   1  -  1;   three  variations. 
[1].     1   +12   +51   +68   +  63   +73   +37;    no  variation. 

Here  three  roots  are  indicated  in  the  interval  [0,  —1],  of 
which  one  must  be  real ;  but,  since  03 2  <  68  x  73,  the  other 
two  must  be  imaginary.  The  equation,  therefore,  has  but  two 
real  roots,  one  in  each  of  the  intervals  [  —  1,  0],   [0,  1]. 

197.  In  the  preceding  examples  the  presence  of  imaginary 
roots,  where  they  occur,  has  been  detected  with  facility,  as  will 
always  be  the  case  except  when  (3,  the  coefficient  of  the  imagi- 
nary sign,  is  so  small  that  the  limits  [A,  ii\  fall  between  con- 
secutive integers. 

When  we  arrive  at  a  doubtful  interval  between  two  consecu- 
tive integers,*  that  is,  one  where  the  coefficients  of  neither 
transformation  give  indications  of  imaginary  roots,  the  roots 
in  the  interval  may  be  (1)  real  roots,  equal  or  nearly  equal,  in 
which  case  we  can,  as  shown  in  (187),  by  the  aid  of  the  proper 
trial  expressions,  determine  them  to  any  degree  of  accuracy 
desired ;  (2)  the  roots,  or  some  even  number  of  them,  may  be 
imaginar}r,  having  (3  small.  Now  the  smaller  (3  is,  the  more 
closely  do  the  imaginary  roots  approach  to  being  equal  real 
roots,  and  the  proper  trial  expressions  will  guide  to  successive 
figures  of  the  real  part  of  the  roots,  till  a  discrepancy  in  the 
suggested  figures  warns  us  that  either  a  group  of  roots  is 
about  to  separate,  or  that  we  have  come  within  the  narrow 
limits  where  the  coefficient  functions  give  indications  of  the 
roots  being  imaginary.  In  the  case  of  a  doubtful  interval, 
therefore,  we  proceed  according  to  the  rules  in  Art.  176,  and 
when  a  discrepancy  in  the  trial  expressions  occurs,  apply  the 
test  for  imaginary  roots. 

*  One  of  which,  obviously,  mast  he  within  .5  of  the  roots,  or  real  part  of  the 
imaginary  roots. 


142  ALGEBRAICAL    EQUATIONS. 

Ex.  1.     Let  the  proposed  equation  be, 
4a;7  _  qxg  _  7^5  +  gz4  +  7^  —  23a:2  —  22a;  —  5  =  0. 

[0].     4-6-7  +  8  +   7  -23  -22  -  5;    three  yar. 
[1].     4   +22   +41   +23  -11   -30  -58  -44;    one  yar. 

Here  two  variations  are  lost ;  and  since  302  <  2(11  x  58 
+  23  x  44),  the  roots  indicated  are  imaginary.  The  remain- 
ing positive  root  is  of  course  real,  and  its  situation  is  easily 
found. 

To  find  the  negative  roots,  changing  the  signs  of  alternate 
terms,  and  proceeding  as  usual,  we  obtain, 

[0].  4  +  6—  7—  8+  7  +  23  —22  +5;  four  var. 
[1].     4+34  +113   +187   +165   +100   +42  +8;  no  var. 

Here  are  four  variations  lost,  and  since  1652  <  2(187  x  100 

—  113  x  42),  two  at  least  of  the  roots  must  be  imaginary. 

Whether  the  other  two  are  so,  evidently  depends  upon  whether 

the  real  root  of  fx  (x)  =  0   in  the  interval  separates  them. 

a      5x2  4K  -,      22 

As    -q5-  =  .4o. .,  and  — — -  =  .47. .,  we  may  expect  the 

roots,  if  real,  to  coincide  in  the  two  figures  .46  at  least; 
proceeding  in  the  approximation,  we  find  the  roots  agree  in 
the  four  figures  .4616,  and  separate  at  the  fifth  figure;  these 
roots  are  therefore  real. 

Ex.  2.    Let  the  proposed  equation  be, 

a*  _  91^4  +  2518a;3  +  17870a;2  +  4617a;  —  82580  =  0. 

Transforming  several  times  by  10,  we  obtain, 

[0].  1-91+2518  +  17870+       4617-       82580;  three  var. 

[10].  1—41-  122  +  48810+  803417+  3458590;  two  var 

[20].  1+9—  762  +  30550  +  1629017  +  15941760;     "     " 

[30].  1+59+   598  +  23090  +  2107417  +  34814930;  no  var. 

By  the  loss  of  a  variation  in  the  interval  [0,  10]  a  real  root 
is  there  indicated ;  the  loss  of  two  variations  in  the  interval 
[20,  30]  indicates  two  roots,  which,  without  subdividing  the 
interval,  we  see  must  be  imaginary,  since  5982  <  23090  x  59. 


FOURIER'S    THEOREM.  143 

To  find  the  situations  of  the  negative  roots,  we  change 
alternate  signs,  and,  seeing  that  8  is  a  superior  limit,  we 
transform  several  times  by  1 ;  thus  we  obtain, 

[0].  1  +   91   +2518  -17870  +   4617  +82580 

[1].  1  +   96   +2892  —  9760  —23200  +71937 

[2].  1  +101   +3286  —     498  —33655  +41966 

[3].  1  +106   +3700  +   9976  -24384  +11201 

[4].  1  +111   +4134  +21722  +   7097  +     600 


two  var 


no  var. 


The  roots  indicated  in  the  interval  [3,  4]  may  be  real,  as 
neither  transformation  has  a  negative  coefficient  function. 
The  roots,  if  real,  are  evidently  nearer  to  4  than  3  ;   as  both 

~600x2  and     ~7097     suggest  -  .16. .,  we  infer  that  the 

7097  21722x2       ^ 

roots  are  not  far  from  3.83...  We  accordingly,  proceeding 
from  the  transformation  by  3,  transform  first  by  .  8,  and  again 
by  .03,  though  there  is  a  slight  discrepancy  in  the  figures 
suggested  by  the  trial  expressions  of  the  transformation  by  .  8, 
the  one  suggesting  .03..,  the  other  .028...  We  obtain  as 
the  three  right-hand  terms  in  this  transformation  19632.7.  ., 
67.76..,  1.058..,  and  as  obviously  67T73  <  19632x2,  the 
roots  indicated  are  imaginary,  the  real  part  being  not  far  from 
3.83..,  and  the  imaginary  part  less  than  .  05  V—  1. 

Ex.  3.     In  the  equation, 
^  +  1755^  +  270450^+14262750^-17791875^  +  5484376=0, 

two  variations  are  lost  in  the  interval  [0,  1],    Here  the  trial 

548  x  2  1779 

expressions  and  — t— — -  both  suggest  .  6.     Proceed- 

J.  {  (  J  -L4:/iO   X  <v 

ing  with  the  approximation,  as  on  the  following  page,  we  find 
the  same  figures  suggested  by  both  expressions  in  each  trans- 
formation, till  we  have  obtained  .6129.  At  the  fourth  trans- 
formation the  trial  expressions  suggest  widely  different  figures, 
and  as  obviously  21982  <  108168  x  147  x  2,  the  roots  are 
imaginary,  having  the  real  part  about  .  6129,  and  the  imag- 
inary part  about  .  0001  V—  1. 

198.  When  the  imaginary  part  is  very  small,  as  in  this 
example,  we  evidently  cannot  obtain  evidence  of  the  imaginary 


144 


ALGEBRAICAL  EQUATIONS. 


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FOURIER'S    THEOREM.  145 

character  of  the  roots  till  we  have  carried  out  the  approxima- 
tion to  so  many  decimals  as  will  make  the  real  part  a  less 
than  (3,  the  coefficient  of  V—  1.  Yet  the  labor  of  ascertain- 
ing that  a  pair  of  roots  is  imaginary  with  a  very  small  imag- 
inary part  is  by  no  means  lost.  It  is  of  great  importance 
to  recognize  such  roots,  as  we  thus  see  that  a  very  slight 
change  in  the  final  coefficient,  or  in  the  conditions  of  the 
problem  that  led  to  the  equation,  would  change  these  imag- 
inary roots  into  equal,  or  nearly  equal,  real  roots.  The  longer 
the  recognition  of  their  imaginary  character  is  delayed,  the 
nearer  do  the  imaginary  roots  approach  to  being  real  roots, 
and  the  more  reason  is  there  to  revise  the  data  in  accordance 
with  which  the  equation  was  framed.  Thus,  in  the  cubic 
equation,  xs  —  nx  —  2.143257  ==  0,  if  we  employ  the  ordi- 
nary approximation  rr  =  3.1416,  we  obtain  three  real  roots, 
two  nearly  equal :  but  if  we  employ  the  closer  approximation 
rr  z=  3.14159,  we  obtain  only  one  real  root,  the  other  two 
being  imaginary  with  (3  comparatively  small. 

199.  In  point  of  facility,  the  method  of  analysis  here  pro- 
posed presents  obvious  advantages  over  that  of  Sturm.  In  the 
latter  method  the  difficulty  of  the  analysis  may  be  said  to 
increase  in  geometrical  ratio  with  the  degree  of  the  equation, 
and  the  calculation  of  the  functions  is  equally  operose,  whether 
the  roots,  from  their  close  proximity,  are  difficult  of  separa- 
tion, or  the  reverse.  In  the  method  here  recommended  the 
labor  of  analysis  may  be  said  to  increase  only  in  arithmetical 
progression  with  the  degree  of  the  equation,  and  is,  compara- 
tively, but  little  affected  by  the  magnitude  of  the  coefficients ; 
and  the  recognition  of  the  character  of  the  roots  in  a  given 
interval  is  delayed  only  in  proportion  to  the  smallness  of  the 
quantity  (real  or  imaginary)  by  which  these  roots  differ. 
There  is  here  no  essential  difference  between  the  analysis  and 
the  solution  of  an  equation,  both  being  effected  by  one  uni- 
form process  consisting  of  a  series  of  transformations  by 
Horner's  Method,  in  which  we  first,  by  transforming  over 
wider  intervals,  ascertain,  by  the  rules  established  by  Fourier's 
Theorem,  in  what  intervals  roots  may  lie ;  then,  by  trans- 
forming over  narrower  intervals,  assisted  by  the  trial  divisors 


14G  ALGEBRAICAL   EQUATIONS. 

(176),  we  either  succeed  in  separating  the  roots,  or  find  by 
the  criterion  established  in  (189)  that  certain  roots  are  imag- 
inary. 

200.  The  calculation  of  imaginary  roots  may,  theoretically, 
be  effected  by  transforming,  by  methods  given  in  Chap.  XII, 
the  equation  in  which  they  occur  into  others  in  which  the  real 
parts,  a  and  (3,  occur  separately  as  real  roots,  which  may  then 
be  obtained  by  methods  already  given.  Practically,  however, 
when  more  than  one,  or  at  most  two,  pairs  of  imaginary  roots 
occur  in  an  equation,  their  calculation  is  very  laborious.  For 
some  of  the  methods  for  determining  such  roots  see  Art. 
212-215. 

EXERCISES. 


Find  the  real  roots  of  the  following  equations : 

1.    a4  +  2a:3  —  12a2  —  x  +  2  =  0. 
x*  +  5a;3  _  15^  _  ^x  —  10  =  0. 
:(a  _  46^2  _  nx  +  2  =  0. 
z*  +  2x*  —  123a:2  +  478a;  —  154  =  0. 
a4  _  12^3  _  9(^2  _  923  +  120  =  0. 

6.  a4  —  16a:2  +  35a:  +  8  =  0. 

7.  ^  _  7^3  +  4a;2  +  20  =  0. 

8.  x5  —  4a4  —  16a3  +  40a2  +  80a:  +  32  =  0. 

9.  x5  —  19a?  +  97a?  +  56a:  —  105  =  0. 

10.  x5  —  32o?  —  181s2  +  5792  =  0. 

11.  x5  —  37o?  +  x2  —  30a;  +  5  =  0. 

12.  x?  —  3663a?  —  75856a:2  +  258962a:  +  792076  =  0. 

13.  a;5  —  6x*  +  a:3  +  13a:2  +  138a:  —  315  =  0. 

14.  x5  —  x*  —  20a:3  +  43a:2  +2x  —  14  =  0. 

15.  a5  +  3a4  —  72a:3  +  4a2  +  1020a:  —  1164  =  0. 

16.  x5  -  152a:3  -  1874a2 +  284848  =  0. 

17.  a6  -  38a-4  -  40a:3  +  217a:2  +  160a:  —  215  =  0. 

18.  a6  —  12a5  +  9a4  +  109a'3  —  34a2  —  267a:  —  26  =  0. 
X*  _  3x5  +  3a4  —  78a?  +  165a:2  —  165a:  +  1265  =  0. 
a*  +  &  +  yx2  _  Qx  +  6  _  0> 


19. 
20 


CUBIC    EQUATIONS.  147 


CHAPTER    XI. 

CUBIC    EQUATIONS. 

201.  A  cubic  equation,  being  of  odd  degree,  has  always 
a  real  root  of  sign  contrary  to  that  of  the  final  term  (11) ;  the 
leading  figure  of  a  root  of  a  cubic  can  thus  be  always  ascer- 
tained by  trying  which  is  the  least  number  that  will  cause  the 
final  term  to  change  sign.  Then,  having  calculated  the  root 
to  which  this  number  is  an  immediately  superior  limit,  we  can 
in  various  ways  determine  the  remaining  roots.  This  manner 
of  proceeding  may,  however,  under  certain  circumstances 
prove  more  laborious  than  is  necessary,  and  the  determination 
of  the  remaining  roots  is  not  so  easily  effected  as  in  the 
method  of  procedure  about  to  be  explained,  by  which  the  ten- 
tative character  of  the  usual  method  is  avoided,  and  the  roots 
obtained  with  a  minimum  of  calculation. 

202.  As  in  Art.  122,  let  the  roots  xu  %2,  %s,  of  x?  +  qx 
-j-  r  =  0,  be  2a,  —  (a-f-/3),  (  —  a— /3),  in  order  of  numerical 
value,  and,  as  before,  writing  the  coefficients  in  terms  of  the 
roots,  we  have, 

tf  _  (3a2  +  (J2)x  _  (2a2  —  2a/32)  =  0. 

Hence  3a2  +  (3*  =  —  q  ; 

4a2  =  —  q,  when  [3  =  a,  ?.  e.,  when  one  root  is  in- 

definitely small, 
and       2a  =  V—q,  the  smallest  possible  value  of  xlf 

the  sign  of  V—  q  being  taken  contrary  to 
that  of  r. 
Again,  as  3a2  +  (32=  —  q, 

3a2         =  —  q,  when  0  =  0,  i.  e.,  when  there  are 
two  equal  roots, 
and  2a  =  V—§q,  the  greatest  possible  value  of  %lf 

when  the  roots  are  all  real. 


148  ALGEBRAICAL    EQUATIONS. 

Again,  as  3a2-f/32  =  —  q, 

3a2— y2  =  —  q,  when  (3  =  y  V—  1?  i-  e.,  when  there 
are  two  imaginary  roots, 
and  2a  >  V  —  iq,   since   3a2  >  —  q.     In  this  last 

case,  if  y2  =  or  >  3a2,  then  q  is  zero  or  positive,  and  we  see 
by  inspection  that  there  are  imaginary  roots. 

Hence,  if  all  the  roots  are  real,  the  numerically  greatest 
must  lie  between  the  limits  V —  q  and  V—£q,  with  the  sign 
of  —  r;  also,  since  the  superior  limit  V  —  \q  is  equal  to  the 
inferior  limit  V-^  q  multiplied  by  Vf,  which  equals  V-ft* 
or  -J  nearly,  the  inferior  limit  differs  from  the  superior  by  less 
than  \  of  the  latter.  Therefore,  if  we  add  one-third  of  itself 
to  —  q,  the  leading  figure  of  the  square  root  of  the  sum  will 
be  either  the  first  figure  of  xx ,  or  may  be  a  unit  too  great,* 
if  the  roots  are  all  real. 

203.   Again,  since 

—  r  =  2a3  —  2a/32, 


.-.     v  — 4r  =  v  8a3— 8a(52  <  2a  when  (3  is  real, 

=  2a  when  j3  =  0, 
>  2a  when  0  =  yV—  1. 

Comparing  these  with  the  results  of  the  preceding  article, 
we  find  that  V  —  iq  >?  =,  or  <  fy — 4r,  according  as  the 
roots  are  all  real  and  unequal,  there  are  two  equal  roots,  or 
there  are  two  imaginary  roots,  so  that  in  every  case  xx  lies 
between  V—  %q  and  V— 4r,  both  having  the  sign  of  the 
latter.     We  are  now  prepared  to  state  the  following  rule  : 

Given  an  equation  z3  +  qx  +  r  =  0,  where  q  is  negative, 
ive  determine  by  inspection  the  leading  figures  of  V—j?q  and 
V—4:r,  talcing  loth  with  the  sign  of  the  latter,  then  (1),  if  the 
first  figure  thus  found  is  greater  than  the  second,  the  roots 
are  all  real  and  unequal  and  the  first  figure  is  either  the 
leading  figure  of  xx  or  may  be  a  unit  too  great ;  (2),  if  the 
first  figure  is  less  than  the  second,  there  are  two  imaginary 

*  In  comparatively  rare  cases  the  leading  fignre  of  V—  $  Q  may  be  greater  than 
the  leading  fignre  of  xx  by  two  units,  as  when  — q  is  close  on  a  square  number,  and 
at  the  same  time  the  equation  has  one  root  very  small  as  compared  with  the  others. 


CUBIC    EQUATIONS.  149 

roots,  and  the  leading  figure  of  xx  is  not  greater  than  the  lead- 
ing figure  of  V— 4r;  (3),  if  both  figures  are  equal,  we  have 
in  each  the  leading  figure  of  xlf  but  the  remaining  roots  are 
doubtful.     See  foot  of  page  59. 

Ex.  1.  a«  —  51a  —  62  =  0. 

Here,  adding  one-third  of  itself  to  51  we  have  68,  and 
62  x  4  =  248 ;  we  see  by  inspection  that  v  68  =  8  +  and 
a/288  =  6  -f,  the  roots  are  therefore  all  real  and  unequal, 
and  the  leading  figure  of  xx  is  8  or  7. 

Ex.  2.  z3  —  6ox  -f  316  =  0. 

Here  V—  $q  =  —  9  +  ,  and  \/—4=r  =  — 10 -f,  there  are 
therefore  two  imaginary  roots,  and  the  leading  figure  of  xx  is 
—  10,  or  —  9. 

Ex.  3.  x^  -  2778z  -  56429  =  0. 

Here  V— $q  =  60  +,  and  fy — 4r  =  60  -f,  the  leading 
figure  of  X\  is  certainly  60  ;  the  remaining  roots  are  doubtful. 
If  it  is  desirable  to  ascertain  their  character  without  solving 
the  equation,  this  may  easily  be  effected  by  comparing  27r2 
and  4^3. 

204.  We  are  in  this  way  able  to  determine  the  leading 
figure  of  Xi  within  very  narrow  limits.  This  root  is  also  the 
most  convenient  for  calculation,  as  there  is  no  other  of  the 
same  sign ;  it  also  differs  from  the  others  by  3a  ±(3,  that  is, 
by  at  least  its  own  value,  so  that  the  trial  divisors  will  be 
found  quite  as  effective  in  guidipg  us  to  the  second  and  suc- 
ceeding figures  of  the  root  as  they  are  in  the  operation  for  the 
cube  root,  which,  as  before  mentioned,  is  but  the  solution  of  a 
cubic  of  the  form  x3  -f-  r  =  0. 

205.  When  xx  has  been  found,  we  may,  if  it  is  commen- 
surable, depress  the  equation  to  a  quadratic  and  thus  find 
x2,  x-3.  In  few  cases,  however,  will  the  root  be  commensur- 
able ;  it  will  therefore  be  found  most  convenient,  after  approxi- 
mating to  Xi  to  six  or  seven  decimals,  to  calculate  the  remain- 
ing roots  by  one  of  the  formulas  deduced  as  follows : 


150  ALGEBKAICAL    EQUATIONS. 


Since  3a2  +  /32  =  —  q,  .-.  (3  =  ±  V—  q  —  3a2,  hence 


-(a±j3)  =  -^±V-q-ix1\ 
Or,  since  2a  (a2— /3s)  =  —  r,  and  3a2-fj32  =  —q,  we  obtain 

=  #2  or  afc. 

Ex.  1.  z3  —  7x  —  7  =  0. 

Here  adding  one-third  of  itself  to  7,  and  multiplying  7  by  4, 
we  obtain  9+  and  28,  the  square  root  of  the  first  and  the 
cube  root  of  the  second  both  begin  with  3,  which  must  there- 
fore be  the  leading  figure  of  xx .  For  comparison  we  place,  on 
the  following  page,  side  by  side,  the  approximation  to  xx  of 
this  equation,  and  the  extraction  of  the  cube  root  of  28.342421, 
which  is  the  cube  of  Xi . 

It  will  be  observed  that  some  modifications  have  been  intro- 
duced into  the  process  for  the  calculation  of  xl}  which  is  thus 
rendered  strictly  analogous  to  that  usually  employed  for  the 
extraction  of  the  cube  root.  Thus,  after  completing  the  right- 
hand  column  by  the  addition  of  three  ciphers,*  we  complete 
the  second  column  by  adding  the  square  of  the  figure  last 
found  to  the  two  rows  of  figures  above,  and  then  appending 
two  ciphers;  the  left-hand  column  we  complete  by  adding 
twice  the  figure  last  found,  and  append  one  cipher.  The 
contraction  is  performed  as  usual,  according  to  the  rules 
in  (170). 

To  find  the  roots  j&2>  xs>  we  shall  employ  the  second  for- 
mula, thus, 


x2  and  xz=  -i(3. 0489173  ±  \Z', 


21 


3.0489173 


=  _i (.30489173  ±  V- 3351253) 
=  —1.692021..  and  —1.356896. 


*  Or  by  bringing  down  three  decimal  figures,  if  there  are  any,  to  the  r:ght  of  the 
integer,  a?  in  the  cube  root  operation. 


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CUBIC    EQUATIONS. 


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152 


ALGEBRAICAL    EQUATIONS. 


Ex.  2. 


a*  —  25z  +  148  =  0. 


Here  25  xf  =  33+,  and  — 148  x4  =  —502  ;  the  square 
root  of  the  first  is  —  5  +,  the  cube  root  of  the  second  is 
—  8  -f- ;  there  are  therefore  imaginary  roots,  and  Cardan's 
Formula  could  be  employed  to  find  xx.  It  is  obvious, however, 
that  the  compact  process  above  exemplified  will  give  the  root 
with  much  less  numerical  labor.  The  leading  figure  of  xY 
must  lie  between  —  5  and  —  8  ;  upon  trial,  it  is  found  to  be 
6,  we  then  proceed  as  follows,  first  changing  the  sign  of  the 
final  term,  so  as  to  make  xx  positive  : 


0 
6 

-25 
36 

—  148     |6.8312004 
66 

6 
12 

11 

36 

-82000 

78432 

180 

8 

8300 
1504 

—3568000 

3429987 

188 
16 

9804 
64 

0 
9 

—138013 
114967 

2040 
3 

113720 

612 

—23046 
22998 

2043 
6 

|20|49 

114332 

114946 

20 

9 
9 

7 
5 

-48 
45 

-3 

114967 

20 

2 

11498 ' 
i 

1 

11|4|9|9|1 


Changing  the  sign  of  the  root  thus  found,  we  have 
Xy  =  —  6.8312004.  The  imaginary  roots,  if  desired,  may  be 
obtained  by  one  of  the  formulae  above  given. 


CUBIC    EQUATIONS.  153 

Ex.  3.  W  —  443a;  —  1396  =  0. 

Here  we  must  divide  the  coefficients  by  7  and  proceed  as  in 

«  •  i        ±u      443       4  j  !396       . 

the  previous  examples;  thus  -=-  x  -q-  =  84  +  ,  and  — =—  x  4 

7  o  / 

=  797+  ;   as  the  square  root  of  84  and  the  cube  root  of  797 
both  begin  with  9,  this  must  be  the  leading  figure  of  xx . 

Ex.  4.  a;3  +  56a;  —  181  =  0. 

Here  we  see  at  once  that  two  roots  are  imaginary,  but  can 

apply  the  rules  for  limits  given  in  (75,  76).     By  the  latter 

181 

-==-  +  1  is  a  superior  limit ;  the  root  proves  to  be  2 .  82 . . . 

o7 

Ex.  5.  a?  —  11a?  -  102a;  —  181  =  0. 

In  order  to  make  the  second  coefficient  divisible  by  3,  we 
multiply  the  roots  by  3   (59) ;  we  thus  obtain, 

ys  _  33?/2  _  918?/  -  4887  =  0. 

We  proceed  to  diminish  the  roots  by  11,  and  obtain, 

z*  +  0  —  12812  —  17647  =  0. 

As  V1281  x  i  and  ^17647  x  4  both  begin  with  40,  we 
proceed  to  calculate  zx  with  40  as  leading  figure  and  obtain 
41 .  3279469.     For  the  remaining  roots  we  have, 


%f  *  =  -  i(41.3279469  ±  ^1281  -  ^^J 

=  —20.6960396,  —20.6319072. 

Adding  11  to  each  of  these  roots,  and  dividing  by  3,  we 
obtain  the  roots  of  the  proposed  equation, 

^  =  17 . 4426489,    x2=  -3. 2106357,    xz  =  —  3 . 2320132. 

EXERCISES. 

Solve  the  following  equations  : 

1.  a*  —  %x  —  1  =  0.  4.    a?  —  Six  —  68  =  0. 

2.  a?  —  63a;  —  9  =  0.  5.    a?  —  7z  +  5  =  0. 

3.  Xs  —  51a;  —  62  =  0.  6.    z3  —  Qx  +  3  =  0. 


154 


ALGEBRAICAL    EQUATIONS. 


7.  38  _  3?z  _  151  =  0. 

8.  Xs  —  9x  +  5  =  0. 

9.  z8  —  782;  —  150  =  0. 

10.  a*  -  Six  +  12  =  0. 

11.  a*  —  43a;  —  4532  =  0. 

12.  V  _  88z  —  251  =  0. 

13.  Xs  —  8x  +  35  =  0. 

14.  3?  _  933  _  15  =  0. 

15.  a?  -  56a;  +  7  =  0. 

16.  a?  —  29a:  +  54  =  0. 

17.  a;3  —  25a;  —  148  =  0. 

18.  &  —  dlx  +  45  =  0. 

19.  Xs  —  15a;  —  2  =  0. 

20.  a?  —  10z  —  13  =  0. 

21.  x*  -  71a;  -  85  =  0. 

22.  a?  -  17a;  -  23  =  0. 

23.  a3  —  23a:  +  35  =  0. 

24.  xs  —  75a;  —  8  =  0. 

25.  x?  —  43a;  —  6  =  0. 

26.  xs  —  93a;  —  51  =  0. 

27.  x*  —  88a;  —  251  =  0. 

28.  x5  —  57z  —  1205  =  0. 


29.  x?  —  65a;  +  987  =  0. 

30.  Xs  —  108a;  —  431  =  0. 

31.  a8  —  132a;  —  443  =  0. 

32.  a*  —  12a;  +  17  =  0. 

33.  x*  —  172a:  +  342  =  0. 

34.  s8  —  149a:  —  530  =  0. 

35.  a?  —  184a;  —  269  =  0. 

36.  a3  —  456a;  +  789  =  0. 

37.  a;3  —  73a:  -  268  =  0. 

38.  a:3  —  345a;  —  678  =  0. 


39. 


234a:  —  567 


0. 


40.  x>  —  91a:  —  331  =  0. 

41.  a;3  —  123a:  +  649  =  0. 

42.  a?  —  123a;  —  456  =  0. 

43.  a,-3  —  678a;  —  901  =  0. 

44.  a?  —  567a;  —  890  =  0. 

45.  a?  —  234a:  —  3634  =  0. 

46.  a*  —  368a:  +  6531  =  0. 

47.  a;3  —  547a:  —  4924  =  0. 

48.  a;3  —  890a;  —  423  =  0. 

49.  a:3  —  496a:  —  33721  =  0. 

50.  a:3  —  432a;  +  3557  =  0. 


51.  a;3  —  20a;2  +  27a;  —  58  =  0. 

52.  a?  _  59^2  +  11353  _  7137  _  0. 

53.  a:3  +  26a:2  +  262a;  +  808  =  0. 

54.  tf  _  49^2  +  658;r  _  1379  =  0. 

55.  438  _  180a;2  +  1896a;  —  457  =  0. 

56.  27a;3  +  27a;2  —  180a;  +  127  =  0. 

57.  512a?  +  192a:2  —  10728a;  +  9409  =  0. 

58.  5a:3  +  124.2a;2  —  338.065a;  —  18606.379  =  0. 

59.  12a;3  -  120a:2  +  326a:  -  127  =  0. 

60.  4a?  -  240a;2  +  3996a;  -  14937  =  0. 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  155 


CHAPTEE    XII. 

SYMMETRICAL    FUNCTIONS   OF   THE    ROOTS. 

206.  A  symmetrical  function  of  the  roots  of  an  equation 
is  an  expression  in  which  the  roots  are  similarly  involved,  so 
that  the  function  is  not  altered  if  any  of  the  roots  be  inter- 
changed. Thus  the  coefficients  of  an  equation  are  symmetrical 
functions  of  the  roots,  since  (45)  in  the  equation 

zn  +  C^x"-1  +  Cn.2xn~2  +  . . . .  CyX  +  <70  =  0 

—  CU     =    «i   +   02   +  «3  + an, 

Cn_2    r=    0102  -T-  0103  +   0203  + 0«-10«> 

—  CU     =     0i0203   +   010204   + 0«-2  0»-10«, 

and  so  on.  These  are  symmetrical  functions ;  since  however 
we  may  interchange  the  roots,  the  function  is  not  altered. 
We  propose  in  the  present  chapter  to  show  that  by  means  of 
the  above  elementary  functions,  given  by  the  coefficients  of 
the  equation,  we  can  obtain  any  rational  symmetrical  function 
of  the  roots  of  an  equation  in  terms  of  its  coefficients. 

20*7.    A  function, 

01"1   +   02™   +    03m  +  •  •  •  •    0nm, 

in  which  each  term  involves  only  one  of  the  roots,  is  said  to  be 
of  the  first  order. 
A  function, 

alma2p  +  0!m03p  +  a2ma^  +  . . . .  0n_iw0^. ., 

which  contains  every  permutation  of  the  roots  taken  two  at  a 
time,  with  m  as  the  exponent  of  the  first  and  p  of  the  second, 
is  said  to  be  of  the  second  order,  and  is  usually  denoted  by 
2  ciima2p,  as  being  the  sum  of  all  the  terms  that  can  be  formed 
like  aima2p. 


156  ALGEBRAICAL    EQUATIONS. 

A  function, 

aFctfa-f  +  axm(h*a£  +  a-Fa^af  +  ...., 

which  contains  every  permutation  of  the  roots  taken  three  at 
a  time,  with  m  as  the  exponent  of  the  first,  p  of  the  second, 
and  q  as  the  third,  is  said  to  be  of  the  third  order,  and  is 
denoted  by  I>aima2pa^q.  For  functions  of  the  fourth  and 
higher  orders  symbols  of  similar  form  may  be  employed. 

The  function  of  the  first  order,  which  is  the  sum  of  the  mth 
powers  of  the  roots,  might  in  like  manner  be  denoted  by 
2  «xm,  but  is  usually  denoted  simply  by  Sm ,  as  in  the  following 
proposition  containing  Newton's  Theorem  for  obtaining  the 
functions  of  the  first  order. 

208.  To  express  the  sum  of  the  mP1  powers  of  the  roots  in 
terms  of  the  coefficients  and  inferior  poivers. 

Let  «i,  a2,  «3, . . . .  an  denote  the  roots  of  the  equation 

f(x)  =  xn  +  CUz""1  +  Cn.2xn~2  + . . . . .  dx  +  C0  =  0, 

also    Si    =  cii   +  a2   +  a%   +  an  > 

S2    =  ai2  +  a22  +  ag  +  a2, 

Sm  =  a{n  +  a2m  +  a3m  +  fl*". 

By  Art.  92  we  have 

J   v  J         x  —  di        x  —  a2  x  —  an 

Each  of  the  divisions  here  indicated  can  be  performed  ex- 
actly by  Art.  1 9,  the  quotient  being  the  equation  depressed  by 
division  by  one  of  its  binomial  factors ;  thus  we  have, 

£&-  =  ar1  +  (a,  +  Cn-i)x-2  +  («i2  +  CU  «,  +  CnJx**  +.... 

+  {a™  +  Cn.i arl  +  Cn_2 am~2  +  . . .  Cn.m)xn-m-' +  . . 

X  W*2 

+  (a2m  4-  CU  a-r1  +  Cn.2  xm~2  +  . . .  G^)z™*  +  . . 

fix)         fix) 
with  similar  results  for   y'v  ;  ,'   y        ,  &c. 
x  —  az     x  —  a± 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  157 

Hence,  by  the  addition  of  these  n  quotients,  we  have 
/,  (x)  =  nx^  +  (Sl  +  nCn.l)x»->+  (S2+  Cll_lS1  +  nCn-2)x»-*+  . . . 

But,  by  Art.  7,  we  have  also, 

f\(x)  =  nxn~l  +  (n—l)Cn_iXn-2  +  (n—2)C„-2xn-3+  .... 

+  (n—m)CH_mxn-m-l-\- 

Equating  the  coefficients  of  corresponding  terms  in  these 
identical  expressions,  we  have, 

^  +  fiCLi  =  (n— 1)CU,  or  Si  +  C„-i  =  0. 

S2  +  CUft  +  »0^  =  (>?-2)6U,  or  £+CU#+20U  =  0. 

4  +  CU&+CUft+»CL*  =  (»-3)CL»,  or 

Ss  +  CL, &  +  Cn_2  Si  +  3  cu  =  o. 

#m  4-  C„-A-i+  CUA-2+  •  •  ^-,-i^+wC,,,  =  (n-m)Cn.m,  or 
4  +  ^-1^-1+^.2^.2+  . .  Cn_m_iSi  +  mCn_m  =  0. 

This  last  formula  gives  Sm ,  the  sum  of  the  mth  powers  of 
the  roots,  in  terms  of  the  coefficients  and  inferior  powers  of 
the  roots.  Thus  having  Si9  we  can  find  S2,  and  then  S3,  and 
so  on,  till  we  reach  Smf  where  m  is  less  than  n. 

The  process  may  be  extended  so  as  to  obtain  Sm ,  where  m 
is  not  less  than  n,  as  follows  : 

Multiplying  the  given  equation  by  xm~n,  we  have 

xm  +  Cn-iXm-1  +  Cn.2xm~2  + C0xm-"  —  0. 

In  this  substituting  in  succession  «l3  a2,  a3, . . . .  an,  for  x, 
and  adding  the  results,  we  obtain 

Sm   +    ^'n-l  Sm-\   +    ^«-2  S,n-2  +    •  •  •  •    ^0  £>m-n  • 

Hence,  making  m  =  ?i,  n  +  1,  w  +  2,  &c,  in  succession, 
we  find,  observing  that  S0  =  «/>  +  a2°  +  . . . .  aH°  =  w, 

«,       +    0.-1  &-,    +    0,-3  S.-2   +    ....    fid    =   0, 

Sn+1  +    Cn-1  #i       +    C«-2  $n-l    + Co  /Si  =    0, 

and  so  on  till  we  obtain  Sm ,  where  m  may  be  of  any  magni- 
tude. 


158  ALGEBRAICAL   EQUATIONS. 

Ex.    To  find  S6,  the  sum  of  the  sixth  powers  of  the  roots  of 
a*  —  2s3  —  5z2  +  7a  —  3  =  0. 

Here  -  Cs  =  2,  -  <72  =  5,  -  Q  =  —  7,  —  G0  =  3. 

#i  =  —  C3  =  2. 

#  =  -  <73#i  -  2C2  =  4  +  10  =  14. 

#3  =  -  C3S2  -  C2Sl  -  3C\  =  28  +  10  -  21  =  17. 

St  =  -ftfls-Q&  — ft/Si— 4Co  =  34  +  70-14  +  12  =  102. 

#5=  _C3^-Cf2^3-C1^2-C0^1  =  204  +  85-98  +  6  =  197. 

S6  =  -  C/3^5-C2/S'4-C1/S'3-(7o/Sr2=394  +  510-119  +  42  =  827. 

Thus  the  sum  of  the  sixth  powers  of  the  roots  is  827  ;  the 

sums  of  higher  powers  can  be  found  by  continuing  the  process. 

To  find  the  sums  of  the  negative  powers  of  the  roots  we  put 

—  for  x,  that  is,  transform  the  equation  into  the  .equation 

involving  the  reciprocals  of  the  roots  (64),  and  apply  the  for- 
mulae as  above. 

209.  A  very  convenient  process  for  finding  the  sums  of 
the  powers  of  the  roots  may  be  deduced  as  follows : 

Since  /!(.)  =  £&-  +  l&L  +  I®-  +  .... 
x  —  ax        x  —  02        x  —  a$ 

a.m      XA  (s)   _       x        ,        a         ,       «?        , 
f{x)      ~  x  —  cii        x  —  a-2        x  —  Oz 

=(-r+(i-r+(-r+- 

=  n  +  S1ar1  +  8zar*+  S3x~s  +  . . . . 

That  is,  if  we  multiply  fx  (x)  by  x,  and  divide  by  f(x)  the 
coefiicients  of  the  quotient  will  be  in  their  order  S0,  Si,  S2, 

Sm .     Thus,  in  the  example  of  the  preceding  article,  if  we 

divide  4^4  —  6z3  —  lOz2  +  Ix  by  x*  —  2^3  —  5x2  +7^  —  3, 
we  obtain  in  succession,  as  the  coefiicients  of  the  quotient, 
4,  2,  14,  17,  102,  &c. 

210.  From  the  equations  in  (208),  expressing  the  sums  of 
the  powers  of  the  roots  in  terms  of  the  coefiicients,  we  can 
obtain  expressions  for  the  coefficients  in  terms  of  Si,  S2,  S3, 
&c. ;  thus, 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  159 

ft-i  =  -ft, 

(7/(.3     =      -  i  (*  +   Gm  &  +    Cn_2  ft), 

CU  =  -  £ (&  +  ft*  ft.!  + . . . .  cn_r_2s,). 

211.  ^4m/  rational  symmetrical  function  of  the  roots  of  an 
equation  can  lc  expressed  in  terms  of  the  coefficients  and 
functions  of  lower  order. 

First,  to  find  the  value  of  2ata2p,  the  function  of  the  sec- 
ond order. 

Since        ft,,  =  at  +  a2m  +  at  +  at  + 

ft,  =  af  +  af  +  af  +  af  +.... 

by  multiplication  we  obtain, 

Sm  ft  =  at+p  +  a2m+p  +  a3m+*  +  «4wl+*  +  . . . . 

+  «tf»  a2*  +  at  a3p  +  ffi™  af  +  a2m  «sp  +  •  •  • . 
that  is, 

ft,  ft  =  Sm+P  +  2  at  a2p,  or  2  Bl-  a2*  =  ft,  ft,  -  Sm+P .    [1]. 

Next,  to  find  the  value  of  2ata2pasq,  the  function  of  the 
third  order,  we  multiply  together  the  equations 

Zataf  =  ata2p  +  at  af  +  at  at  +  a2maf  +  .... 
ft  =  a?        +  af        +  a£        +  af       +-... 

and  obtain  a  result  consisting  of  three  partial  products  : 

(1)  the  sum  of  the  products  of  the  form 

af+qaf  —  Zat+qa2p, 

(2)  the  sum  of  the  products  of  the  form 

at+qa2,n  =  2at+9a2m, 

(3)  the  sum  of  the  products  of  the  form 

at  a  fag  =  lata-faf; 
thus, 

ft  x  lataf  =  Zat+qaf  +  Zat+qa2m  +  ^atafag. 

Substituting  for  lataf,  Zat+qaf,  and  laf^a-t  their 
values  obtained  by  formula  [1],  we  have 

2  at<hva<£  =  Sm  ft  ft  —  Sm+q  ft  —  Sp+q  Sm  +  %Sm+p+q .     [2]. 


160  ALGEBRAICAL    EQUATIONS. 

By  proceeding  in  a  similar  manner  we  can  obtain  formula? 
for  functions  of  the  fourth  and  higher  orders  expressed  in 
terms  of  the  sums  of  the  powers  of  the  roots ;  and  as  these 
(208)  can  be  expressed  by  integral  functions  of  the  coefficients, 

any  rational  symmetrical  function    la{na2p amr  can  be 

expressed  by  integral  functions  of  the  coefficients. 

These  are  called  the  elementary  symmetrical  functions; 
and,  as  it  is  by  the  combination  of  these  that  every  complex, 
rational,  and  symmetrical  function  is  formed,  it  follows  that 
the  value  of  every  rational  symmetrical  function  of  the  roots 
can  be  expressed  by  the  coefficients  of  the  equation. 

212.  The  formulae  obtained  above,  where  the  exponents 
m,  p,  q,  &c,  were  assumed  to  be  unequal,  must  be  modified 
if  we  suppose  any  of  the  exponents  to  be  equal.  Thus,  if 
m  =  p  in  the  formula 

lafaf  =  SmSp  —  Sm+p, 

since  then  a{*a2p  =  axp  a2m,  the  terms  in  1  (iima2p  become  equal, 
two  and  two,  and  2aima2p  becomes  22(a1a2)ni,  therefore,  in 
this  case, 

2a1ma2m  =  i(SJ-S2m). 

Similarly,  if,  in  Za^a^a^,  m  =  p,  we  have 

2  K  <h)m<hq  =  i  (SJ  Sq  -  S2m  Sq  -  2Sm+q  Sm  +  ^S2m+q) ; 

and  if  m  =  p  =  q,  I>a]ma2mazm  becomes  2'3l(aia2ai)m;  or 

Z{aia2chy  =  i  (SJ  -  SS2m Sm  +  2S3m). 

213.  By  means  of  the  theory  of  symmetrical  functions  we 
are  able  to  transform  an  equation  into  another  the  roots  of 
which  shall  be  given  functions  of  those  of  the  proposed  equa- 
tion. The  following  transformation,  having,  at  one  time,  been 
proposed  as  a  mean  of  separating  the  roots  of  an  equation,  is 
of  some  interest. 

214.  To  transform  an  equation  into  another  whose  roots 
are  the  squares  of  the  differences  of  the  roots  of  the  proposed 
equation. 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  161 

Let  aly  a2,  a3, an,  be  the  roots  of  the  proposed  equa- 
tion, then  the  roots  of  the  required  transformed  equation 
will  be 

(a,  —  a2)\    {fr  —  a3)2, (a-2  —  a6f,  &c. 

and  will  be  in  number  \n  (n— 1),  the  possible  number  of  dif- 
ferent combinations  of  n  things  taken  two  at  a  time.     The 
degree  of  the  transformed  equation  will  therefore  be  \n  (n— 1) 
=  m  suppose. 
Let  ym  +  qiym~l  +  q2ym~2  +  . . . .  qm  =  0 

be  the  transformed  equation,  and  slr  s2,....sr  denote  the 

sums  of  the  first,  second, rth  powers  of  its  roots.    When 

we  have  determined  the  values  of  these  sums  of  the  powers 
we  can  obtain  the  coefficients,  since  (210)  qx  =  —  Si , 
q2  =  —  \  (s2  4-  q\  Si),  &c. ;  it  remains  therefore  to  find  a  gen- 
eral expression  for  any  sum  as  sr . 

Let   Si,   S2, Sm9  denote,  as  before,  the  sums  of  the 

powers  of  the  roots  of  the  proposed  equation,  then 

(x  _  aiyr  +  (x-  a2)2r  +  (x-  a,)2r  +  . . .  ■  (x  -  an)2r 

=  nx2'- -  2rS1x2r~1  +  2r(f'~1) S2x2'~2  +  ....S2r. 

In  the  first  member  of  the  above  equation,  when  we  put  ax 
for  x,  we  obtain  the  sum  of  the  rth  powers  of  the  squares  of  the 
differences  in  which  a\  comes  first ;  when  we  put  a2  for  x,  we 
obtain  the  sum  of  the  rth  powers  of  the  squares  of  the  differ- 
ences in  which  a2  comes  first,  and  so  on ;  if  therefore  we 
put  cii ,  a2 ,  a3, . . . .  an,  in  succession  for  x,  and  add  all  the 
results,  we  obtain 

2sr  =  nS2r  -  2rS,  #2,_i  +  ^p^  S2  S,r.2  -....*&. 

The  terms  to  the  right  hand  that  are  equidistant  from  the 
beginning  and  end  are  equal;  collecting  and  dividing  by  2, 
we  have 

sr  =  nS2r  -  2rSl 82r-i  +  ^^ S2 S2r.2  +  . . . . 


+  i(_l)^^-l)^..(r+_l)^ 


162  ALGEBRAICAL  EQUATIONS. 

To  obtain  the  required  transformation,  we  accordingly  first 
find  Sl9  S2,  $,,....  by  (208)  from  the  coefficients  of  the 
proposed  equation ;   we  then,  by  means  of  the  formula  just 

given,  find  in  succession  su  s2, sm ;  and,  finally,  by  means 

of  the  formulas  qx  =  —  su  $2  =  —  ifa  +  qisj,  &c,  obtain 
the  coefficients  of  the  required  equation. 

The  impracticable  nature  of  a  method  of  analysis  requiring, 
as  a  mere  preliminary  to  the  separation  of  the  roots  of  an 
equation  of  the  sixth  degree,  the  calculation  of  fifteen  coeffi- 
cients through  these  three  stages,  is  self-evident. 

215.  The  theory  of  symmetrical  functions  may  also  be 
applied  to  the  elimination  of  one  of  the  unknown  quantities 
between  two  simultaneous  equations.    See  Art.  227. 

216.  The  sums  of  the  powers  of  the  roots  may  be  advan- 
tageously employed  in  certain  cases  to  obtain  an  approximation 
to  the  roots  of  an  equation.  * 

Let  Oi,  a2,  (hi denote  the  roots  of  an  equation  in  de- 
scending order  of  magnitude,  then  we  have 

Sm+i         aim+1  +  d2m+1  +  «3W+1  4-  •  •  •  • 


Sm         a{n     +a2m     +agm     + 


»+er+(sr  + 


Now  the  fractions   l—J     ,    (—  )  ,  &c,  may  be  made  as 

small  as  we  like  by  taking  m  large  enough.     -~^  is  therefore 

an  approximation  to  the  greatest  real  root  ax ,  provided  that 
it  be  greater  than  the  modulus  of  any  imaginary  pair,  the 
approximation  becoming  closer  as  m  increases. 

217.    But  if  there  be  a  pair  of  imaginary  roots  whose 
modulus  is  greater  than  the  greatest  real  root,  we  cannot  ob- 

*  This  was  first  suggested  by  Newton,  and  was  further  developed  by  Lagrange. 
The  subject  will  be  found  very  fully  treated  in  Murphy's  Equations. 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  163 

tain  an  approximation  to  that  root  as  above.  For  (Trig.)  these 
imaginary  roots  may  be  put  under  the  form  |u(cos  0  ±  sin  0a/-1), 
where  \i  is  the  modulus ;  then 

-  2cos(m  +  l)0+  (^)      +  (^T\  .... 

-  r 


& 


»«"•     +(?)    +  (?)' 


#m+i  -5"  $»  therefore  approximates  to  ju  ■ —  ^     ,  which 

may  have  any  value. 

218.   Again,  if  the  two  greatest  roots,  aY,  a2,  are  real, 
if  «x  and  «2  are  imaginary,  then 

In  either  case,  therefore,  we  have 

#OT    =  «!w    +  fl2m ,    nearly,  when  m  is  taken  large  enough. 
Sm+1=a1m+1+a2m+1,  nearly, 
^m+2  =  «iwl+2+  «2W+2,  nearly, 

each  equation  being  more  nearly  true  than  the  preceding  one ; 

•••     Sm    Sm+2—Sm+i2  =  («i«2)m    («i— a2)2  =  um, 

Sm+i  SM+3  —  Sm+22  =  («l«2),n+1(«i— «2)2  =  «m+i5 
.-.    — —  =  axa2,  approximatively. 

That  is,  if  from  every  three  terms  of  the  series  /Si,  82,  S3,  &c, 
another  series  2  wm  be  formed  by  subtracting  the  square  of  the 
means  from  the  product  of  the  extremes,  the  quotients  ob- 
tained by  dividing  each  term  of  this  new  series  by  the  term 
that  precedes  it,  approximate  more  and  more  nearly  to  the 
product  of  the  two  greatest  roots.  If  the  two  greatest  roots 
are  real,  then  the  greater  being  known  by  (216),  the  second 
becomes  known  by  this  process.  If  they  are  imaginary,  we 
can  proceed  to  find  their  sum  as  follows. 


164  ALGEBEAICAL  EQUATIONS. 

219.  As  in  the  preceding  case,  we  can  find  from  the  values 
of  Sm,  Sm+U  &h-2,  &c,  that 

SMSm+3  —  Sm+iSm+2  =  ffra2m(«i+a2)(«i-«2)2  nearly,  =  vm  say. 

Dividing  this  by  um ,  we  obtain  ax  -f  a2  nearly ;  that  is,  if 

from  every  four  terms  of  the  series  Sl9  S2,  Ss, another 

series  2  vm  be  formed  by  subtracting  the  product  of  the  means 
from  the  product  of  the  extremes,  then  the  quotients  obtained 
by  dividing  each  term  of  this  series  by  the  corresponding  term 
of  the  series  2  um  approach  more  and  more  nearly  to  the  sum 
of  the  two  greatest  roots. 

The  sum  and  the  product  of  two  imaginary  roots  having 
been  found  in  this  way,  each  can  then  be  determined  by  a 
quadratic. 

Ex.  a4  —  xs  +  4z2  +  x  —4  =  0. 

28m=  1,-7,-14,  29,96,-34,-503,-347,  2083,3838,-6159. 
2  um=  -63,-399,-2185,-10202,-49444,-241211,-1168158. 
2  ^=-69,-266,-2308,-11323,-50414,-245363,-1207713. 

The  first  series  being  divergent  shows  that  the  roots 
«!,  a2  are  imaginary.     From  the  second  series  we   obtain 

aiCk  =  =  4.84;    from  the  second  and  third  we 

241211       1207713 
have    ai  -f-  «2  =    TTfi81^8   =   1-03;    the  roots  alf  a2  are, 

therefore,  \  (1 .  03  ±  4 . 3  V^l). 

For  the  purpose  of  calculating  real  roots  this  method  is 
obviously  too  laborious  if  we  aim  at  accuracy  to  several  places 
of  decimals ;  but  in  some  cases  this  appears  to  be  the  most 
convenient  process  yet  proposed  for  approximating  to  the  real 
and  imaginary  parts  of  impossible  roots. 

220.  If  an  equation  of  any  degree  has  only  two  imaginary 
roots,  they  can  easily  be  determined,  if  required,  in  the  follow- 
ing manner,  which  may  be  advantageously  employed  to  deter- 
mine the  remaining  roots  of  any  equation  when  all  but  two 
have  been  calculated. 

Let  kx  and  Km  denote  respectively  the  sum  and  product  of 
the  ascertained  values  of  all  the  roots  but  two,  and  ti  and  t2 


SYMMETRICAL    FUNCTIONS    OF    THE    ROOTS.  165 

the  sum  and  product  of  the  remaining  two ;  then  ( 45 ) 
tx  =  —(Cn.x  +  Kx),  and  t2  =  {—l)*C0-7-Km  ;  the  two  roots,  there- 
fore, can  be  obtained  from  the  formula,  x  =  \  (^liV^i2 — 4£2). 

Ex.  1.     In  the  equation  already  given, 

ic4  —  x3  +  4z2  +  x  —  4  =  0, 

we  see  at  once  (189),  since  1  —  4x2  is  negative,  that  there 
are  two  imaginary  roots,  and  the  final  sign  being  negative,  the 
other  two  are  real,  one  of  each  sign.  By  Horner's  Method 
the  values  of  these  roots,  .892232..  and  —.923262..,  are 
calculated  accurately  to  six  places  of  decimals  with  less  labor 
than  is  required  to  obtain  the  first  series  2Sm.  The  sum  of 
these  roots  is  —.03103,  therefore  tx  =  1.03103;  the  product 
of  the  above  roots  is  —  .823765  ;  dividing  the  final  term  —  4 
by  this,  we  obtain  t2  =  4.855762. . ;  hence  we  obtain  for  the 
roots  |(1.03108..  ±  4.284857.  ,'V^l).  One  advantage  in 
obtaining  the  last  two  roots  in  this  manner  is  that  we  are 
enabled  to  perform  much  of  the  calculation  by  logarithms. 

Ex.  2.     In  the  equation, 

ofi  +  173^  +  2356a?  -f  10468a:2  —  14101a;  +  4183  =  0, 

there  is  a  pair  of  imaginary  roots.  The  sum  of  the  real 
roots  is  found  to  be  —157.438702..,  hence  tx  =  (—173 
+  157.438702)  =  -  15.561297. ..  The  product  of  the  real 
roots  is  —  49 .  9555 . . ;  dividing  —  4183  by  this,  we  obtain 
t2  =  83.73454..;  hence  we  find  the  roots  to  be  —7.780648 
±  V— 23.19607. 

221.  If  all  the  roots  of  an  equation  of  the  fourth  degree 
are  imaginary,  we  can  always  obtain  their  values  by  means  of 
Descartes'  reducing  cubic  (129).  If,  then,  an  equation  of  any 
degree  has  only  four  imaginary  roots,  we  can,  after  obtaining 
the  real  roots,  depress  it  to  a  biquadratic,  and  then  obtain  the 
imaginary  roots.  Unless,  therefore,  an  equation  has  as  many 
as  six  imaginary  roots,  we  can  depress  the  equation  to  one 
capable  of  algebraical  solution,  and  thus  obtain  the  values  of 
the  imaginary  roots  with  comparative  facility.  But  if  an 
equation  of  the  sixth  degree,  or  one  that  has  been  depressed  to 
that  degree,  has  all  its  roots  imaginary,  the  method  of  Art.  219 


166  ALGEBRAICAL    EQUATIONS. 

appears  to  afford  the  readiest  solution.  We  can  first,  by  that 
method,  find  the  two  greatest  of  these  roots ;  then,  by  taking 
the  coefficients  in  reverse  order,  and  repeating  the  process,  we 
obtain  the  reciprocals  of  the  two  smallest  roots;  finally,  having 
thus  determined  four  roots,  we  can  easily  obtain  the  sum  and 
product  of  the  remaining  pair,  and  thus  determine  them  also. 

222.  Theoretically  the  determination  of  the  values  of 
imaginary  roots  may  be  effected  as  follows.  If  in  the  proposed 
equation  f(x)  =  0,  we  substitute  a  +  <3  V— 1  for  x,  we  ob- 
tain (26)  a  result  of  the  form  P  +  Q  V^l  =  0 ;  hence  we 
obtain  two  equations,  P  =  0,  and  Q  =  0,  each  involving  a 
and  (3.  As  will  be  shown  in  the  next  chapter,  we  can  from 
these  equations  always  determine  the  corresponding  values  of 
a  and  0.  But,  if  f(x)  be  of  the  sixth  degree,  this  method 
requires  the  formation  and  part  solution  of  an  auxiliary  equa- 
tion of  the  fifteenth  degree,  while,  if  f(x)  is  of  the  eighth  de- 
gree, the  auxiliary  equation  rises  to  the  28th  degree. 

EXERCISES. 

Find  the  values  of  S2 ,  Sz , . . .  S6 ,  in  the  following  equations : 

1.  x*  —  5a;2  +  6x—  1  =  0.         4.    ^  —  3.^  —  7^  +  5  =  0. 

2.  <c3  +  12a;  +  8  =  0.  5.    x5  +  Gx*  +  5xi—x  +  2  =  0. 

3.  xi  +  2x*— 3z2— 4z  +  l  =  0.    6.   a;5  —  Gx*  —  10  =  0. 

Find  the  remaining  two  roots  of  the  following  equations,  the 
sums  %,  and  products  nn_2,  of  the  other  roots  being  given  : 

1.  x*  -  8a;3  +  16a?  —  x  —  2  =  0 ; 

«!  =  7.864328,  n,  =  13.41444. 

2.  a,4  +  8a?  +  51a;2  +  30a;  +  30  =  0 ; 

k,  =  —12.830412,  k2  =  5.142960. 

3.  a;4  -  19a;2  -  20a;  +  5  =  0 ; 

«!  =  1.173254,  k2  =  17.335048. 

4.  x5  —  x*  —  11a?  —  3a;2  +  23a;  +  13  =  0 ; 

«,  =  4.678823,  *3  =  —3.678823. 

5.  a*  -  3a4  —  10a?  -  11a;2  +  121a;  +  70  =  0  ; 

k,  =  7.073375,  %  =  —8.146750. 
0.    x«  -  19a4  —  22a?  +  84a;2  +  179a;  +  85  =  0; 

*!  =  4.027525,  ka  =  -20.366725. 


ELIMINATIONS.  167 


CHAPTER    XIII. 

ELIMINATION. 

223.  Two  equations  involving  each  two  unknown  quan- 
tities x  and  y  may  be  denoted  by  F(x,  y)  =  0,  and  f(x,  y) 
=  0 ;  their  solution  consists  in  the  determination  of  the  sys- 
tems of  values  of  x  and  y  which  cause  both  equations  to 
vanish  simultaneously.  In  certain  cases  the  solution  is  readily 
effected.  Thus,  if  from  one  of  the  equations  we  are  able  to 
obtain  an  expression  for  the  value  of  one  of  the  unknown 
quantities,  as  x,  in  terms  of  the  other  y,  by  the  substitution 
of  this  value  for  x  in  the  other  equation  we  obtain  an  equa- 
tion involving  y  only,  the  roots  of  which  can  be  determined 
by  methods  already  given ,  and  these  values  of  y  substituted 
in  the  expression  for  x  in  terms  of  y,  will  give  the  corres- 
ponding values  of  x. 

Ex.         x3  —  (y  +  2)a2  +  (if  —  2y)x  —  12  =  0, 
x2  —  yx  +  2  =0. 

From  the  second  equation  we  obtain  y  =  x  -\ .     This 

x 

value  of  y  substituted  in  the  first  equation  gives  us 

x*  —  2x*  -f  4z3  —  4:X  —  8  =  0. 

Of  this,  one  value  is  x  =  2,  and  for  this  value  of  x,  y  —  3, 
which  values  cause  both  equations  to  vanish  simultaneously. 
The  other  values  of  x,  being  determined,  would  in  like  manner 
give  corresponding  values  of  y. 

224.  Also,  if  the  first  members  of  the  equations  can  be 
readily  resolved  into  factors,  the  solution  of  the  equations  may 
be  made  to  depend  upon  that  of  equations  of  inferior  degrees. 
If,  for  example,  we  find  F(x,  y)  =  UV  U"  =  0,  and  f{x,  y) 
=  VV  =  0,  then  the  systems  of  values  that  satisfy  the  pro- 


1C8  ALGEBRAICAL    EQUATIONS. 

posed  equations  may  be  obtained  by  solving  the  simultaneous 
equations  U  =  0  and  V  =  0  ;  U  =  0  and  V  =  0  ;  U'  =  0 
and  V  =  0 ;  U'  ==  0  and  F'  =  0 ;  C/"=0  and  7=0; 
Z7"  =  0  and  V  =  0. 
Let  F(x,  y)  =  x*y  —  (f—2y)x2+(yz  +  2y)x  -if-  2y2  =  0, 
f(x,y)  =  a?  -  (y  +  1)  z2  -  (2  -  y) »  +  2  =  0. 

These  may  be  resolved  as  follows, 

F(x,y)  =  (o;2-^  +  2/2)(^-^_2)  =  0, 
/(»,y)  =  (x*-xy-2){x  +  y  -  1)   =0. 

The  systems  of  values  that  satisfy  the  proposed  equations 
can  therefore  be  obtained  by  solving  the  equations 

x2  —  xy  +  if  =  0  I ,         x2  —  xy  —  y%  =  0  > 
x2  —  Xy  —  2  =  0  )  ,        x  +  #  '—  1   =  0  >  ' 

^-^-2=0),        ^_2/3_2=0) 
z2_a^_2=0J\        a;  4.  ^  _i=o  J. 

225.  But  if  one  of  the  factors  of  F(x,  y)  is  identical  with 
one  of  the  factors  of  f(x,  y) ;  if,  for  example,  U  and  V  are 
identical,  then,  obviously,  any  values  of  x  and  y  that  make 
U  vanish  will  cause  both  the  proposed  equations  to  vanish 
simultaneously.  If  U  involves  both  x  and  y,  we  can  assign 
any  value  we  please  to  one  of  these  unknowns,  and  determine 
the  corresponding  value,  or  values,  of  the  other  that  will  cause 
U  to  vanish,  and  thus  obtain  as  many  solutions  as  we  please. 
If  U  involves  only  one  of  the  unknown  quantities,  we  can 
satisfy  the  equations  F(x,  y)  =  0,  and  f(x,  y)  =  0  by  giving 
to  that  unknown  quantity  any  value  that  causes  U  to  vanish, 
and  to  the  other  any  value  we  please.  In  order,  therefore,  that 
the  proposed  equations  be  determinate,  that  is,  be  satisfied 
by  a  limited  number  of  corresponding  values  of  x  and  y,  they 
must  not  have  a  common  divisor  involving  x  or  y. 

226.  To  eliminate  between  two  equations,  each  involving 
the  same  two  unknown  quantities,  is  to  deduce  an  equation, 
involving  only  one  of  these  unknown  quantities,  the  solution 
of  which  will  furnish  all  the  values  of  that  quantity,  which, 
taken  with  the  corresponding  values  of  the  other,  satisfy  the 


ELIMINATION-.  169 

proposed  equations.     The  equation  thus  deduced  is  called  the 
final  equation,  and  its  roots  are  called  suitable  values. 

227.  To  eliminate  one  of  the  unknown  quantities  between 
two  equations  involving  tivo  unknown  quantities  by  means  of 
symmetrical  functions. 

Let        p0xn  +  pi  xn~l  +  p2xn~2  +....pn  =  0,  [1] 

q0x>»  +  qxxm~l  +  q2xm~2  +  . . . .  qm  =  0,  [2] 

be  the  proposed  equations,  in  which  the  coefficients  pQ,  pi} 

p2, q0,  ql9  q29 are  rational  integral  functions  of  y. 

Assume  that  we  can  solve  equation  [1]  in  respect  to  x9  in 
terms  of  y,  and  that  these  values  are  a,  b,  c,  &c.  By  substi- 
tuting these  values  in  equation  [2],  we  obtain  n  equations 
involving  only  y,  namely : 

qQam  +  qxam-1  +  q2am~2  +  . . ..  qm  =  0  \ 

qQbm  +  q,bm~l  +  q2bm~2  +  . . . .  qm  =  0   >       [3]. 

qQcm  +  qicm~l  +  q2cm~2  +  . . -. .  qm  =  0  J 

In  general  the  solution  of  [1]  cannot  be  effected ;  but  by 
multiplying  together  all  the  above  n  equations,  we  obtain  a 
final  equation  which  has  for  roots  all  the  suitable  values  of  y. 
For  this  equation  will  vanish  for  any  of  the  values  of  y  that 
makes  any  of  its  factors  vanish,  and  for  no  others ;  and  any  of 
these  is  a  suitable  value.  For  suppose  the  first  of  the  equa- 
tions [3]  vanishes  for  y  =  (3,  and  that  when  (3  is  put  for  y  in 
the  function  a,  the  value  is  a ;  then  x  =  a,  y  =  p  will 
satisfy  the  two  proposed  equations.  Now  in  the  final  equa- 
tion, which  is  the  product  of  all  the  equations  [3],  the  factors 
only  change  places  when  we  interchange  any  of  the  quantities 

a,  b,  c, ,  thus  the  product  is  a  symmetrical  function  of 

these  quantities,  which   may  be  expressed  in  terms  of  the 

coefficients  pQ,  pl9  p2, of  equation  [1],  and  thus  we  can 

obtain  the  final  equation  in  y  which  contains  all  the  suitable 
values,  and  no  other. 

228.  Though  the  preceding  method  of  elimination  has  the 
merit  of  furnishing  a  final  equation  with  all  the  suitable  values, 


170  ALGEBRAICAL    EQUATIONS. 

and  no  others,  the  calculations  required  are  so  tedious  that 
the  method  is  not  generally  available.  Upon  it  is  based,  how- 
ever, the  following  theorem  : 

229.  The  degree  of  the  final  equation  resulting  from  the 
elimination  of  one  of  the  unknown  quantities  between  two  equa- 
tions of  the  mth  and  nth  degrees  respectively,  ivill  not  exceed  mn. 

Let  p^xn  +  pxx"-1  +  p2xn-2  + pn  =  0, 

q0xm  +  qixm~l  +  q-2xm-2  +  . . . .  qm  =  0, 

be  the  proposed  equations  in  which  the  coefficients  are  func- 
tions of  y.  Here  it  is  supposed  that  the  sum  of  the  exponents 
of  x  and  y  does  not  exceed  n  in  any  term  of  the  first  equa- 
tion, or  m  in  the  second,  so  that  a  coefficient  pr  or  qr  is  a 
function  of  y  not  higher  than  the  rth  degree. 

Suppose  x  to  be  eliminated  by  the  method  of  Art.  228 ; 
then  the  first  member  of  the  final  equation  in  y  consists  of  a 
series  of  terms,  each  of  which  is  the  product  of  n  factors,  and 

is  of  the  form  qram-r  x  qs  bm~s  x  qtC"-*  x Since  (227)  the 

series  of  terms  forms  a  symmetrical  function  of  a,  b,  c, , 

the  aggregate  of  the  terms,  with  exponents  as  above,  is, 

qr  qs  qt I  am~r  bm~s  cm-1 

Now  the  degree  of  y  in    qrq8qt....   is  not  higher  than 

r  +  s  -f  t  -f ,  and  it  will  be  shown  that  the  degree  of  y 

in  1  am~r  bm~s  c m~l . . .  cannot  exceed  m—  r+m— s+m — 1+ '. ., 
so  that  the  degree  of  the  whole  product  is  not  greater  than  mn. 
For  from  the  formulae  in  Art.  208  we  see  that  Sp ,  for  example, 
does  not  contain  any  power  of  y  greater  than  yp ;   and  from 

the  formulae  in  Art.  211  a  function  lahbkcl will  contain 

powers  and  products  of  Si,  S2, Sh+M.l+..,  in  each  term 

the  sum  of  the  letters  subscript  to  S  being  h  +£  +  /+  . .  ;  the 
degree  of  y  in  the  function  1  am~r  bm~s  c111'1 . . .  cannot  therefore 
exceed  m+r+m—s+m—t-\- . . . ;  hence  no  power  of  y  in 
the  final  equation  can  be  higher  than  y"m. 

230.  Although  the  degree  of  y  in  the  final  equation  can- 
not exceed  mn,  it  may  in  certain  cases  be  less.  By  an  exten- 
sion of  the  process  to  any  number  of  equations  we  are  led  to 
the  general  theorem  discovered  by  Bezout,  namely : 


ELIMINATION.  171 

If  between  any  number  of  equations  involving  the  same 
number  of  unknown  quantities  we  eliminate  all  but  one,  the 
degree  of  the  final  equation  will  not  exceed  the  product  of 
the  degrees  of  the  original  equations. 

231.  The  elimination  of  one  of  the  unknown  quantities 
between  two  equations  is  most  conveniently  effected  by  the 
following  method,  based  upon  the  operation  for  obtaining  the 
greatest  common  measure  of  two  algebraical  quantities. 

To  determine  the  systems  of  values  that  will  satisfy  two 
equations  involving  two  unknown  quantities. 

Let  F(x,  y)  =  0  and  f(x.,y)  =  0  be  two  equations,  which 
we  shall  suppose  have  no  common  factor  and  thus  admit  of  a 
limited  number  of  solutions.  Suppose  that  x  =  a  and  y  =  (3 
are  values  that  satisfy  these  equations,  then  F(a,  y)  =  0  and 
f(a,  y)  =  0  are  both  satisfied  by  y  =  (3.  F(a,  y)  and  f(a,  y) 
must  therefore  have  a  common  measure  which,  equated  to  zero, 
will  have  for  a  root  a  value  of  y  which,  conjointly  with  x  =  a, 
will  satisfy  the  proposed  equations.  Having  therefore  arranged 
both  equations  according  to  descending  powers  of  x,  we  pro- 
ceed by  the  operation  for  the  G.  C.  M.  till  we  arrive  at  a 
remainder  independent  of  x,  say  r/>(?/). 

Now  if  no  factors,  functions  of  y,  have  been  introduced  or 
suppressed  in  the  operation,  so  as  to  avoid  having  y  in  a  denom- 
inator, (p(y)  =  0  will  comprise  all  the  suitable  values  of  y,  and 
no  others.  For  unless  (f>(y)  =  0,  F(a,  y),  and  f(a,  y)  have  no 
common  measure  and  therefore  do  not  vanish  simultaneously. 
But  if  it  has  been  necessary  to  introduce  factors,  functions  of 
y,  then  some  of  the  roots  of  <j>(y)  =  0  may  not  be  suitable 
values,  having  been  introduced  with  these  factors;  and  if 
factors  have  been  suppressed,  there  may  be  suitable  values  of 
y  not  found  among  the  roots  of  <f>(y)  =  0. 

232.  A  process  by  which  the  suitable  values  of  y  may  be 
determined  is  furnished  in  the  following  method  due  to  M.  M. 
Labatie  and  Sarrus. 

Let  ^4  =  0  and  B  =  0  be  the  two  simultaneous  equations, 
of  which  neither  has  a  factor  a  function  of  y  only,  and  B  is 
not  of  higher  dimensions  in  x  than  A.     Let  c  denote  the 


172  ALGEBRAICAL    EQUATIONS. 

factor  by  which  A  must  be  multiplied  to  make  it  divisible  by 
B ;  let  q  be  the  quotient,  and  rR  the  remainder,  where  r  is 
a  function  of  y  only.  Let  cx  denote  the  factor  by  which  B 
must  be  multiplied  to  make  it  divisible  by  R,  let  qx  be  the 
quotient,  and  rY  Rx  the  remainder,  where  ?\  is  a  function  of  y 
only.  Suppose  that,  proceeding  in  this  way,  at  the  third  di- 
vision we  arrive  at  a  remainder  r2  independent  of  x.  Thus 
we  have  the  identities, 

cA  =  qB  +  rR,    \ 

ClB  =  q1R  +  r1Rl,  V  [1]. 

c2R  =  q2R\+  r2.       ) 

Let  d  be  the  G.  C.  M.  of  c  and  r,    dx  the  G.  C.  M.  of 

-j-1  and  ri,   cl2  the  G.  C.  M.  of  -j-j-  and  r2.     We  have  now 

cl  Cl  Cl\ 

to  show  that  the  solution  of  the  equations  A  =  0,  B  =  0, 
will  be  obtained  by  solving  the  equations  : 


B  =  0  J  ,        i?    =  0  J  ,        i?!  =  0 


I.   -4W  the  solutions  obtained  from  this  system  of  equations 
satisfy  the  equations  A  =  0,  B  =  0. 

Dividing  both  members  of  the  first  identity  [1]  by  d,  we 
have 

ia  = 15  +  ;*       p» 

Since  *7  is  the  G.  C.  M.  of  c  and  r,  -7  and  —  are  integral 

~  a  a 

functions  of  y,  therefore  also  -^B  is  an  integral  function ;  but, 

by  hypothesis,  B  has  no  factor  which  is  a  function  of  y  only, 
therefore  d  must  divide  q. 

From  [3]  we  see  that  the  values  of  x  and  y  that  satisfy  the 

7*  C  1* 

equations  -=  =  0  and  B  =  0  make  -^^4  also  vanish ;  but  -r 
^     c        d  d  a 

and  -7  have  no  common  factor,  therefore  these  values  make  A 
d  r 

vanish.    Hence  all  the  solutions  of  -=•  =  0  and  B  =  0,  satisfy 
A  =  0,  B  =  0. 


divides  —±  and  rl9  and  does  not  divide  R.     Divide  by  dly 
then  putting  M  for  -^  and  Ml  for  CirJ~7q9l9  we  have 


ELIMINATION.  173 

Again  multiplying  both  members  of  the  identity  [3]  by  c{ , 
and  substituting  for  cxB  its  equivalent  from  the  second  iden- 
tity in  [1],  we  have 

d  ad 

The  expression   - — j  is  integral,  since  r  and  q  are 

divisible  by  d;   the  expression  is  also  divisible  by  dx,  for  ^ 

CCi 

d 

nr,       TUT    ?M     

d  ddi 

MA  =  *«  +  lMR-  [4] 

Multiplying  both  members  of  the  second  identity  [1]  by  -^ 

we  have  c«_B  =  e^R       c 

d  d  d 

Since  dx  divides  C-~  and  ru  it  will  divide  —  ^-R;  but  R 
is  not  divisible  by  dx ,  therefore  —y-  must  be  so.  Dividing  by 
dx  and  putting  N  for  y  and  JVi  for  -—-,  we  have 

The  identities  [4]  and  [5]  show  that  all  the  values  of  z  and 

?/  that  make  -^  and  i?  vanish  make  -r\A  and  -r^-i?  vanish, 
J  dx  ddi  ddx 

but  -=-4-  and  -^   have  no  common  factor;    therefore  all  the 
a  dx  d\ 

solutions  of  the  equations   -y =  0   and   i2  =  0    satisfy  the 
equations  A  =  0,  B  =  0. 

In  the  same  way,  if  we  multiply  both  members  of  the  iden- 
ties  [4]  and  [5]  by  c2,  and  substitute  for  c2R  its  equivalent 
from  the  third  identity  [1],  we  obtain 

C2-A  =  M2Rl+^-Mly  [6]. 


ddxd2  d> 

CClC2-B  =  A^  +  ^ZVi,  [7]- 


ddid2  d2 


174  ALGEBRAICAL    EQUATIONS. 

where  M2  and  JV2  are  integral  functions  of  x  and  y.     From 
these  identities  we  see  that  all  the  solutions  of  the  equations 

-j-  =  0  and  Mi  =  0,  satisfy  the  equations  A  =  0,  B  =  0. 

II.  ^4ZZ  #ie  tioZuflg  o/  a;  «tzc?  y  that  satisfy  the  equations 
A  =  0,  B  =  0  are  included  among  the  solutions  obtained 
from  the  systems  of  equations  [2]. 

The  identity  [3]  may  be  written 

NA  -  MB  =  4s.  [8]. 

Multiply  [4]  by  B,  [5]  by  J,  and  subtract,  then 
(MYB  -  NXA)R  +  (MB  -  iVJ)^1 2?!  =  0, 
therefore,  by  [8], 

(MXB  -  NiA)R  -  ?~RRi  =  0, 

therefore  Mx  B  -  NXA      =  ~  Rx.  [9] . 

In  the  same  way,  from  [6]  and  [7],  we  may  deduce 

M2B-N2A     =  ^fj-R2  [10]. 

a  a  i  a2 

This  last  identity  shows  that  all  the  values  of  x  and  y  that 

rnr2 

ddid2 


make  A   and  B  vanish  make     ,  !  I    vanish.      Hence  the 


equations 


L  -  o    -1  -  o    -2  -  o 
d  ~  u'    dl  ~  u'    d2  ~  u' 


supply  all  the  suitable  values  of  y. 

Suppose  that  x  =  a,  y  =  (3  are  values  that  satisfy  the  equa- 
tions A  =  0,  B  =  0,  then 

T 

(1).  If  (3  is  a  root  of  the  equation  —  =  0,  then  the  values 

a  j, 

x  =  a,   y  =  (3    evidently  satisfy  the   equations   -=■  =  0    and 

B  =  0. 

(2).  If  (3  is  not  a  root  of  -^  =  0,  but  is  of  -y  =  0,  then. 
r  cl  Cli 

since  -j  does  not  vanish  when  y  =  (3,  it  follows  from  [8]  that 
a 


7*9 

y  =  |3  make  i?x  vanish ;  thus  they  satisfy  the  equations  —  =  0 


ELIMINATION".  1 75 

the  values  x  =  a,  y  =  (3,  make  i?  vanish  ;   thus  they  satisfy 
5  =  0  and  7£  =  0. 

(3).  If  (3  is  not  a  root  of  the  equation  -=-  =  0,  nor  of  -r  =  0, 
but  is  of  the  equation  —  =  0,  then,  since  -rT  does  not 
vanish  when  y  =  j3,  it  follows  from  [9]  that  the  values  x  —  a, 
y  =  (3  make 
and  Ei  =  0. 

7*  7*i  7*o 

The  equation    7  7   7  =  0,  which  furnishes  all  the  suitable 
^  6?  di  a2 

values  of  y,  may  be  called  the  final  equation  in  y. 

233.  If,  instead  of  a  final  remainder  in  y,  we  arrive  at  a 
remainder  zero,  this  shows  that  A  and  B  have  the  preceding 
remainder,  say  i?b  as  a  common  factor.  Then,  as  we  saw  in 
Art.  225,  the  equations  A  =  0  and  B  =  0  will  be  satisfied 
by  an  unlimited  number  of  values  of  x  and  y  derived  from 
the  equation  Rx  =  0.     By  dividing  by  this  common  measure, 

A  B  A 

we  obtain  -^-  and  -^j-,  which  have  no  common  factor,  then  -^  =  0 

and  -=-  =  0  will  be  satisfied  for  a  limited  number  of  values 
Mi 

of  x  and  y,  which  may  be  found  as  above. 

If  the  final  remainder  is  a  mere  number,  say  /,  then  the 
equation  I  =  0  being  absurd  shows  that  the  proposed  equa- 
tions are  incompatible  with  each  other. 

Ex.  1.    a*3  4-  3yx*  +  (Sy*  —  y  +  1)»  +  yz  —  y2  +  2y  =  0. 
a*2  +  2y:c  +   y*  —  y  —  0. 

Here  the  first  remainder  is  #  +  2y,  so  that  r  =  1 ;  the 
second  remainder  is  ?/2  —  y,  which  is  independent  of  x.    All 

the  solutions  are  furnished  by  -j  =  0  and  R  =  0,  that  is,  by 

the  equations    y2  —  y  =  0,   and  a;  +  2#  =  0. 

Ex.  2.      x*  +  2?/e2  +  2y(y—2)x  +  ?/2  —  4  =  0. 
z2  +  fyz  +  2t/2  —  5#  +  2  =  0. 

The  first  remainder  is  (y— 2)(a*  +  ?/  +  2) ;  so  that  7*  =  y— 2, 


176  ALGEBRAICAL  EQUATIONS. 

R  =  x  +  y  -f  2  ;  the  second  remainder  is  y2  —  by  +  6,  which. 

T 

is  independent  of  #.    All  the  solutions  are  furnished  by  -  =  0 

and  B  =  0,  that  is,  by  the  equations  #  —  2  =  0  and  x2  +  2xy 

+  2y2  —  by  +  2  =  0;  and  by  -^  =  0  and  R  =  0,  that  is, 

by  x2  —  by  +  6  =  0    and   a  +  #  +  2  =  0.     The  final  equa- 
tion in  y  is  (y  —  2)  (?/2  —  5y  +  6)  =  0. 

Ex.  3.    x*  +  3ijx2— 3x2-{-3y2x—6yx— x—tf>— 3y2—y-\-3  =  0. 
xz—3yx2-\-3x2  +  3y2x—6yx—x—ys  +  3y2  +  y—3  =  0. 

The  first  remainder  is  2{y —  1)  (3x2 -\- y2 —  2y —  3);  the 
second  remainder  is  8  (y2  —  2y)x  ;  the  third,  which  is  inde- 
pendent of  x,  is  y2  —  2y  —  3.  The  solutions  are  furnished 
by  the  systems  of  equations, 

y— 1  =  0  and  x*—  3yx2  +  3x2  +  3y2x-6yx-x-y3+3y2+y-3  =  0; 
y2  —  2y  =  0  and  3x2  +  y2  —  2y  —  3  =  0  ; 
2/2  _  9y  —  3  =  0  and  x  =  0. 

The  final  equation  in  y  is  (y—l)(y2—2y)(y2—2y—3)  =  0. 

Ex.  4.  yx?  —  (y3  —  3y  —  l)x  +  y  =  0. 

x2  —   y2  +  3  =0. 

The  first  remainder  is  z  -f-  ?/ ;  the  second  is  3 ;  the  proposed 
equations  are  therefore  incompatible. 

exercises. 
Solve  the  following  equations  : 

1.  xs  _  x2  +  {2y2  —  3±)x  —  2y2  +  34  =  0  \ 
x2  —  {y  +  rt)x  —  2if  +  ±y  =  Q  ,    )" 

2.  (y  —  l)x2  +  yX  +  y2  —  2y  =  0  \ 
{y  -  l)x  +  y  =  0  }  ' 

3.  a*  +  (8?/  -  13)z  -f  ^  —  7#  -f  12  =  0  ") 


a;2  —  (±y  +    1)#  +  y2  +  5?/  =  0 

(y  -  l)x*  +  y(y  +  l)z2  +  (3?/2  +  ; 

(y  -  l)x2  +  y{y  +  l)s  +   3#2  -         =0 

l)x2  —  2x-\-by 
yx2  —  bx  +  4# 


4.  (y  -  l)a*  +  y(y  +  l)x>  +  (3?/2  +  y  -  2)x  +  Sty  =  0  ) 

5.  (y  —  2)x2  —  2x  +  by  —  2  =  0  ) 

0         )  ' 


ANSWERS 


177 


ANSWERS 


Page  4. 

1.  /(O)  =-13,  /(l)  =  -  7,   /(2)  =       71,  /(3)  =       299. 

2.  /(O)  =     18,   /(l)  =     26,  /(2)  =     546,  /(3)  =     7416. 

3.  /(O)  =-16,  /(l)  =-51,  /(2)  =-268,  /(3)  =  -  511. 

4.  /(O)  =       4,   /(l)  =  -  7,  /(2)  =-528,  /(3)  =-7331. 

5.  fr(x)  =  n(n—l) (n—r  +  l)Cnxn-r  +  (^—1)  (w— 2) 

(w— /,)(7,(_ia;,l-'-14- [rtf,. 


Page  15. 


1.  /(3)  = 

2.  7(4)  = 

3.   /(5)   = 


184. 

198. 

1646. 


4.  /(-2) 

5.  /(H) 


1449. 

85814289. 


Page  2H. 


1.  f(x)  =  (a: -3)  (a: -3)  (a: -2)  (a: +  7). 

2.  f(x)  =  (a;  -  8)  (a; -5)  (a; +  4)  (a; +  2)  (a; +  1). 

3.  /(a;)  =  x3  —  93a;  +  308  =  0. 

4.  f(x)  =  x*  —  12a:3  +  49a:2  —  78a:  +  40  =  0. 

5.  f(x)  =  a^  +  4a:3  -  79a:2  —  106a:  +  840  =  0. 

6.  f{x)  ==  x5  —  25a:4  +  220a:3  —  832a:2  +  1480a;  —  1600  =  0. 


1.  x  = 

2.  a?  = 

3.  a;  = 

4.  a:  = 

5.  x  == 


3  -  V- 
—  5  — 

i(7  +  V-=3i), 
Hi- V-3), 

4  -  V7,     -  2, 


Page  28. 

3,     -  3  ±  Vl2. 
/"-l,    i(5±V29). 


i 


(5  ±  V73). 


i(l±  V-14). 
-  6. 


K8  ALGEBRAICAL    EQUATIONS. 


Page  33. 

1.    x  =  3. 

4.    a;  =  2,  5, 

2.    x  =  5. 

5.    a  =  6. 

3.    x  =  4. 

Page  37. 

1.  a6  —  W  -f  3a  +  5  =  0.  5.  ?/12  +  ?/+  y5—  ?/4— ^  =  0. 

2.  a5  +  7a2  —  x  —  2  =  0.  6.  9a10-a9-a6-  6a5  +  1  =  0. 

3.  7?/39  +  3?/36-5?/32-7  _  o.  7.  (1  -  x)  (1  +  a)3  =  0. 

4.  2?/®  f-  ^  +  3  =  0. 

Page  39. 

1.  ^  _  3^2  _  55^  _  500  =  0. 

2.  if  —  54?/2  +  1470  =  0. 

3.  if  -  2357?/2  +  367000?/  -  745000  =  0. 

4.  */4  +  2625?/2  -  154350?/  +  22509375  =  0. 

Page  41. 

6.    The  roots  occur  in  pairs  differing  only  in  sign,   thus 
f(—x)  =0  is  identical  with  /(a)  =  0. 

Page  42. 

1.  llif  —  20?/2  +  by  —  3  =  0. 

2.  15?/4  +  72?/3  +  54?/2  -7  =  0. 

3.  23?/5  -  7?/4  -  32?/3  +  17?/  -  1  =  0. 

Page  44. 

1.  Xs  -  145a6  —  22a4  —  156a2  +  25  =  0. 

2.  25a8  —  170a6  +  479a4  -  1175a2  +  361  =  0. 

3.  xia  _  54^.3  +  qzW  _  6156^4  +  5796a;2  +  6£5  _  q. 

Page  47. 

1.  ?/4  +  5?/3  -  2?/2  +  817?/  +  4050  =  0. 

2.  #4  +  84?/3  +  332?/2  +  573?/  +  327  =  0. 

3.  ll?/4  +  147?/3  +  708?/2  +  1480?/  +  1191  =  0. 

4.  3?/5  +  47?/4  +  313?/3  +  1068#2  +  1780?/  +  1043  =  0. 

5.  8?/5  -  200?/4  +  2000?/3  —  9922?/2  +  24113?/  -  23055  =  0. 


ANSWERS.  179 

Page  48. 

1.  if  -  33?/  -  71  =  0. 

2.  y*  -  275?/  4-  1692  =  0. 

3.  if  +  186?/  +  1807  =  0. 

4.  ^  _  174^a  _  1261?/  _  2549  =  0. 

5.  y*  —  726^  -  6616?/  +  6045  =  0. 

Page  53. 

1.  Superior  limit  2,  inferior  limit  —  15. 

2.  Superior  limit  4,  inferior  limit  —  6. 

3.  Superior  limit  5,  no  negative  root. 

4.  Superior  limit  4,  inferior  limit  —  5. 

Page  64. 

1.  f{x)  =  (x-2Y(x-S)  =  0. 

2.  f{x)  =  (s -f)2  (a +  6)  =  0. 

3.  f(x)   =  (x  —  o)2  (a«  +  10a;  +  3)  =  0. 

4.  f(x)   =  (x  -  J)2(232  -  x  -  1)   =  0. 

5.  f(x)  =  (32  — 23  — 1)2(3  +  5)   =  0. 

6.  f(x)   =  (x*  +  33  +  1)2(.32-  63  +  12)   =  0. 

Page  67. 
The  reduced  equations  are  : 

1.  3?/2  —  1y  +  25  =  0.  4.    2?/2  -  11?/  -  2  =  0. 

2.  5?/2  +  Sx  —  66  =  0.  5.    #a  —  23#  —  1  =  0. 

3.  f-  +  13?/  —  40  =  0. 

Page  74. 

1.  3=  a/5,    -i(l±  a/^)a/5. 

2.  3  =  ±  J  (V2  ±  v^JV?. 

3.  3  =  a/7,     i  (V5  -  1  ±  V- 10  _.2Vt)ff7, 

-i(V5  +  l±  V_  10  4- 2^/5)^7. 

4.  3  =  ±  S/2,     }(1  ±  V~3)v%     -i(l  ±  V^V^ 


0. 


x  =     i(Vo  -  1  ±  V-  10-2  V5), 
-  i  (a/5  +  1  ±  V_10  +  2  a/5). 
Multiply  the  roots  in  (5)  by  ±  a/3. 


180  algebraical  equation's. 

Page  74. 

7.  The  roots  are  all  the  products,  with  changed  signs,  of 

the  roots  of  x5  —  1  =  0  and  xz  —  1  =  0. 

8.  Multiply  the  roots  in  (7)  by  —  1#20. 
The  depressed  equations  are  : 

1.  ^3    _|_    y2    _    %y    _    1     —     0> 

2.  y4-y3  —  3y*  +  2y  —  l=:  0. 

3.  ^s  _  yi  _  4^3  +  3^2  +  3^  _  i  —  o. 

4.  ^4  +  ^2  +   1    _   0. 

Page  82. 
The  real  roots  are : 

1.  x  =  —.32748...  4.    x  =  -.696549... 

2.  x  =  4.107243. ..  5.    a  =  2.576222. .. 

3.  a;  =  .896922...  6.    re  =  — 2.896025. .. 

Page  86. 

1.  x  =^6,     -  2,     -|(3  ±  a/21). 

2.  x  =  i(l±V-H,     -_Hl±V-3). 

3.  cr  =  2,     —  4,     1  ±  2  V—  L_ 

4.  x  =  i(7  ±  a/37^  i(5  ±  Vl7). 

5.  a;  =  tV(3  ±  V69),     -  |(5  ±  a/^). 

Page  95. 

The  depressed  equations  are  : 

1.    x2  +  x  =   ±  (2*  +  23). 

**-¥•  =  ±(t-+4 

3.  a?  —  4z  +  2  =   ±  5. 

4.  *  _  IQx  +  1*  =   ±  *J. 

5.  cc8  —  —  x  +  5  =   ±  -jr-  x. 

31 


6. 

2            3 

X1  —  —x   = 

± 

(!- 

7. 

£3  +  12^2  _ 

a;  - 

5 

~  2    ":    : 

8. 

7?  +  3a;2   = 

± 

(4a;  +  5), 

±  (n.  +  | > 


1. 


ANSWERS.  181 

Page  105. 

[4,  5],  two  imaginary  roots. 

2.  [6,  7],  two  imaginary  roots. 

3.  [7,  8],  two  imaginary  roots. 

4.  [_.6,  -.5],    [-.5,  -.4],    [2,3],    [4,5]. 

5.  Roots  all  imaginary. 

6.  [0,  1],  four  imaginary  roots. 

Page  125. 

I.  x  =  .6458252,  .6458290,  -52.6458252,  -100.6458290. 
2  x  =  .4142852,  .4142843,  -72.4142852,  -142.4142843. 
3.  x=  .7321401,  .7321410,-114.7321401,-170.7321410. 
4  x  =  .8285606,  .8285659,  -74.8285606,  -144.8285659. 

5.  x  =  .4499705,  .4499761,  -84.4499705,  -102.4499761. 

6.  x  =  .2360680,  .2360663,  —4.2360680,  two  imag.  roots. 

7.  x—  .1414284,  .1414246,  —14.1414284,  two  imag.  roots, 

8.  x=  .2396769,  .2396118,  -54.2396769,  two  imag. roots. 

9.  x=  .4422711,  .4422496, —90,4422711,  two  imag.  roots. 
10.  x=  .5612971,  .5612977, -158,5612971,  two  imag.  roots, 

n  Page  146. 

1.  x  =  2.618034,   .381966,  -4.561552,  -.438447. 

2.  x  =v5,  3.938004/- -1-1&3492,  -3.804512.' 

3.  x  =  6.854102,   .145898,  -.298438,  -6.701562. 

4.  x  =  5.645751,  5.643651,  .354249,  —13.643651. 

5.  x  =  17.440307,   .732051,  -2.732051,  -3.440307. 

6.  x=  —.208712,  —4.791288,  two  imag.  roots. 

7.  x  =  6.282681,  1.883579,  two  imag.  roots. 

8.  x  =  4.82843,  3.75877,  -.69461,  -.82843,  -3.06418. 

9.  x  —  .700263,  —2.41742,  —14.58258,  two  imag.  roots. 
10.  x  =  5.656854,  5.656653,  —5.656854,  two  imag.  roots. 

II.  x  —  4.132933,   .162278,  —6.162278,  two  imag.  roots. 

12.  x  =  68.63515,  4.63519,  -2,  -30.63519,  -40.63515. 

13.  x  =  3,  3,  4.073375,  two  imaginary  roots. 

14.  x  =  2.87939,  2.87298,  -.53209,  -.65271,  -4.87298. 

15.  x  =  4.73265,  4.73205,  1.267949,  -4.38548,  -9.34717. 

16.  x  =  12.328868,  12.328828,  —12.328828,  two  imag.  roots. 


182 


algebeaical  equations. 

Page  146. 

78281,    -1.47848,    -2.94883, 


17.  x  =  G.18658,    2.16601, 

-4.6081. 

18.  x  =  10.09902,  3.67882,  2,  —.90098,  two  imag.  roots. 

19.  x  =  3.80295,  3.80204,  four  imag.  roots. 

20.  x  =  No  real  roots. 


Pagi 

i  152. 

1. 

X 

= 

1.879385, 

—  1.53209.* 

2. 

X 

= 

8.007741, 

-  7.864837. 

3. 

X 

— 

7.685514, 

-6.431126. 

4. 

X 

— 

7.73265, 

-6.347169. 

5. 

X 

■=. 

2.166011, 

—  2.948828. 

6. 

X 

= 

2.145102, 

-2.669079. 

7. 

X 

= 

7.549859, 

two  imaginary  roots. 

8. 

X 

= 

2.669443, 

-3.246542. 

9. 

X 

= 

9.670143, 

—  7.639755. 

10. 

X 

= 

9.257633, 

—  9.395601. 

11. 

X 

= 

17.414059, 

two  imaginary  roots. 

12. 

X 

= 

10.570921, 

—  7.332814. 

13. 

X 

= 

—  4.073375,    two  imaginary  roots. 

14. 

X 

= 

9.723306, 

-  9.562042. 

15. 

X 

= 

7.420014, 

—  7.545049. 

16. 

X 

= 

3.886699, 

-6.146939. 

17. 

X 

= 

6.8312004, 

two  imaginary  roots. 

18. 

X 

= 

4.608100, 

—  6.186583. 

19. 

X 

= 

3.938004, 

—  3.804512. 

20. 

X 

= 

3.678823, 

two  imaginary  roots. 

21. 

X 

— 

8.970896, 

—  7.747852. 

22. 

X 

=: 

4.681165, 

—  3.09226. 

23. 

X 

= 

3.668800, 

-5.426743. 

24. 

X 

zzz 

8.713102, 

-  8.606420. 

25. 

X 

= 

6.626123, 

—  6.486525. 

26. 

X 

zzz 

9.906962, 

—  9.356784. 

27. 

X 

= 

10.570917, 

—  7.332828. 

28. 

X 

= 

12.412788, 

two  imaginary  roots. 

29. 

X 

= 

—  12.105179,  two  imaginary  roots. 

*  The  remaining  root  is  easily  found  by  subtraction. 


ANSWERS.  183 

Page  152. 

30.  x   =  11.996912,  -  6. 231475. 

31.  x  =   12.807578,  -9.139393. 

32.  x  =  —4.027525,    two  imaginary  roots. 

33.  x  —  11.976841,    —14.014402. 

34.  x  =   13.G99870,     -9.719021. 

35.  x  =  14.243780,    -12.704220. 
30.  x  =  20.429925,    -22.171735. 

37.  x  =  9.991179,    two  imaginary  roots. 

38.  x   =  19.488225,  -17.500218. 

39.  x  =   10.388155,  —13.902291. 

40.  x  —  11.003075,    two  imaginary  roots. 

41.  q:  —    —13.131002,    two  imaginary  roots. 

42.  x  =  12  015331,    —8.215575. 

43.  x  =  20.079053,    —25.340054. 

44.  x  =  24.560824,    —22.984299. 

45.  x  —  20.318709.    two  imaginary  roots. 

46.  x  =   —25.070094,    two  imaginary  roots. 

47.  x  =  27.000090,    -13.558083. 

48.  x  =  30.007726,    —29.592320. 

49.  x  —  37.388585,    two  imaginary  roots. 

50.  x  =   —24.077597,   two  imaginary  roots. 

51.  x  —  18.7234001,    two  imaginary  roots. 

52.  x  =  25.190234,    18.470444. 

53.  x  =   —5.312233,    two  imaginary  roots. 

54.  x  =  23.213112,    23.229537. 

55.  x  =  28.521277,    10.231939. 
50.  x  =  1.358088,    1.023502. 

57.  x  =  3.837334,     .931099. 

58.  x  =  1 1 .  197334,    two  imaginary  roots. 

59.  x  —    .405001,    two  imaginary  roots. 

60.  x  =  33.521277,    21.231939. 

Page  160. 

The  required  sums  of  the  roots  are  : 

1.  13,  38,  117,  370,  1180.  4.  9,  48,  145,  483,  1740. 

2.  -24,-24,288,480,-3204.     5.  -12,  -15,  76,  140,  -393. 

3.  io,  -14,  40,  -92,  256.         0.  0,  18,  0,  50,  108. 


184  algebraical   equations. 

Page  166. 
The  remaining  roots  of  the  proposed  equations  are : 

1.  .438447,    —.3027756 

2.  2.416199,    2.414214. 

3.  .208712,    —1.381966. 

4.  -  J- (3. 67882  ±  V— .601169). 

5.  _  1(4.07338  ±  V-  17.7715). 

6.  -  i  (4.02753  ±  V—  .  662873). 

Page  175. 
The  solutions  are  given  by  the  equations  : 

1.  x2  +  ty2  —  34  =  0  I      x2  +  2y2  —  34  =  0  ) 
x  +  y    —    7  =  0  {'    x  —2y  =  0  j 

x  —  1  =  0  \         x  —  1     =   0   \ 

z  +  y—<?  =  o  )'       x  —  %y  =   0  J " 

2.  f  —  2y  =  0    and    (y  —  1)  x  +  y  =  0. 

3.  #  —  1  =  0  > 

x2  —  (4#  +  1)  x  +  2/2  -f-  5#  =  0    )  ' 

a;  —  1    =    0  ) 

CC2  _  (4^  +  i)  x  _j_  ^2  +  5^  _  o    F ' 

4.  ^  _  1  —  0    and    (#  —  1)  a?  +  2y  =  0. 

5.  (3#  —  10)  2  +  if  +  6y  =  0  | 
y*  +  12?/3  +  87?/a  -  200?/  +  100  =  0  >  ' 


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